Calculus Examples

$y = | x^{2} - 6 x |$

Step 1

Write $y = | x^{2} - 6 x |$ as a function.

$f (x) = | x^{2} - 6 x |$

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - 6 x$ .

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 6 x$ .

$\frac{d}{d u} [| u |] \frac{d}{d x} [x^{2} - 6 x]$

Step 2.1.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$\frac{u}{| u |} \frac{d}{d x} [x^{2} - 6 x]$

Step 2.1.3

Replace all occurrences of $u$ with $x^{2} - 6 x$ .

$\frac{x^{2} - 6 x}{| x^{2} - 6 x |} \frac{d}{d x} [x^{2} - 6 x]$

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

By the Sum Rule, the derivative of $x^{2} - 6 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x]$ .

$\frac{x^{2} - 6 x}{| x^{2} - 6 x |} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x])$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{x^{2} - 6 x}{| x^{2} - 6 x |} (2 x + \frac{d}{d x} [- 6 x])$

Step 2.2.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{x^{2} - 6 x}{| x^{2} - 6 x |} (2 x - 6 \frac{d}{d x} [x])$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{x^{2} - 6 x}{| x^{2} - 6 x |} (2 x - 6 \cdot 1)$

Step 2.2.5

Multiply $- 6$ by $1$ .

$\frac{x^{2} - 6 x}{| x^{2} - 6 x |} (2 x - 6)$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Reorder the factors of $\frac{x^{2} - 6 x}{| x^{2} - 6 x |} (2 x - 6)$ .

$(2 x - 6) \frac{x^{2} - 6 x}{| x^{2} - 6 x |}$

Step 2.3.2

Factor $x$ out of $x^{2} - 6 x$ .

Tap for more steps...

Step 2.3.2.1

Factor $x$ out of $x^{2}$ .

$(2 x - 6) \frac{x \cdot x - 6 x}{| x^{2} - 6 x |}$

Step 2.3.2.2

Factor $x$ out of $- 6 x$ .

$(2 x - 6) \frac{x \cdot x + x \cdot - 6}{| x^{2} - 6 x |}$

Step 2.3.2.3

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$(2 x - 6) \frac{x (x - 6)}{| x^{2} - 6 x |}$

Step 2.3.3

Simplify the denominator.

Tap for more steps...

Step 2.3.3.1

Factor $x$ out of $x^{2} - 6 x$ .

Tap for more steps...

Step 2.3.3.1.1

Factor $x$ out of $x^{2}$ .

$(2 x - 6) \frac{x (x - 6)}{| x \cdot x - 6 x |}$

Step 2.3.3.1.2

Factor $x$ out of $- 6 x$ .

$(2 x - 6) \frac{x (x - 6)}{| x \cdot x + x \cdot - 6 |}$

Step 2.3.3.1.3

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$(2 x - 6) \frac{x (x - 6)}{| x (x - 6) |}$

Step 2.3.3.2

Apply the distributive property.

$(2 x - 6) \frac{x (x - 6)}{| x \cdot x + x \cdot - 6 |}$

Step 2.3.3.3

Multiply $x$ by $x$ .

$(2 x - 6) \frac{x (x - 6)}{| x^{2} + x \cdot - 6 |}$

Step 2.3.3.4

Move $- 6$ to the left of $x$ .

$(2 x - 6) \frac{x (x - 6)}{| x^{2} - 6 \cdot x |}$

Step 2.3.3.5

Factor $x$ out of $x^{2} - 6 x$ .

Tap for more steps...

Step 2.3.3.5.1

Factor $x$ out of $x^{2}$ .

$(2 x - 6) \frac{x (x - 6)}{| x \cdot x - 6 x |}$

Step 2.3.3.5.2

Factor $x$ out of $- 6 x$ .

$(2 x - 6) \frac{x (x - 6)}{| x \cdot x + x \cdot - 6 |}$

Step 2.3.3.5.3

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$(2 x - 6) \frac{x (x - 6)}{| x (x - 6) |}$

Step 2.3.4

Multiply $2 x - 6$ by $\frac{x (x - 6)}{| x (x - 6) |}$ .

$\frac{(2 x - 6) (x (x - 6))}{| x (x - 6) |}$

Step 2.3.5

Factor $2$ out of $2 x - 6$ .

Tap for more steps...

Step 2.3.5.1

Factor $2$ out of $2 x$ .

$\frac{(2 (x) - 6) x (x - 6)}{| x (x - 6) |}$

Step 2.3.5.2

Factor $2$ out of $- 6$ .

$\frac{(2 x + 2 \cdot - 3) x (x - 6)}{| x (x - 6) |}$

Step 2.3.5.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$\frac{2 (x - 3) x (x - 6)}{| x (x - 6) |}$

Step 2.3.6

Reorder factors in $\frac{2 (x - 3) x (x - 6)}{| x (x - 6) |}$ .

$\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$

Step 3

Find the second derivative of the function.

Tap for more steps...

Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{(x (x - 3)) (x - 6)}{| x (x - 6) |}]$ .

$f'' (x) = 2 \frac{d}{d x} (\frac{(x (x - 3)) (x - 6)}{| x (x - 6) |})$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x (x - 3) (x - 6)$ and $g (x) = | x (x - 6) |$ .

$f'' (x) = 2 (\frac{| x (x - 6) | \frac{d}{d x} (x (x - 3) (x - 6)) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x (x - 3)$ and $g (x) = x - 6$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) \frac{d}{d x} (x - 6) + (x - 6) \frac{d}{d x} (x (x - 3))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.4

Differentiate.

Tap for more steps...

Step 3.4.1

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) (\frac{d}{d x} (x) + \frac{d}{d x} (- 6)) + (x - 6) \frac{d}{d x} (x (x - 3))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) (1 + \frac{d}{d x} (- 6)) + (x - 6) \frac{d}{d x} (x (x - 3))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.4.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) (1 + 0) + (x - 6) \frac{d}{d x} (x (x - 3))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.4.4

Simplify the expression.

Tap for more steps...

Step 3.4.4.1

Add $1$ and $0$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) \cdot 1 + (x - 6) \frac{d}{d x} (x (x - 3))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.4.4.2

Multiply $x$ by $1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) \frac{d}{d x} (x (x - 3))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.5

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x - 3$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (x \frac{d}{d x} (x - 3) + (x - 3) \frac{d}{d x} (x))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.6

Differentiate.

Tap for more steps...

Step 3.6.1

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (x (\frac{d}{d x} (x) + \frac{d}{d x} (- 3)) + (x - 3) \frac{d}{d x} (x))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.6.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (x (1 + \frac{d}{d x} (- 3)) + (x - 3) \frac{d}{d x} (x))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.6.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (x (1 + 0) + (x - 3) \frac{d}{d x} (x))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.6.4

Simplify the expression.

Tap for more steps...

Step 3.6.4.1

Add $1$ and $0$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (x \cdot 1 + (x - 3) \frac{d}{d x} (x))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.6.4.2

Multiply $x$ by $1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (x + (x - 3) \frac{d}{d x} (x))) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.6.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (x + (x - 3) \cdot 1)) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.6.6

Simplify by adding terms.

Tap for more steps...

Step 3.6.6.1

Multiply $x - 3$ by $1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (x + x - 3)) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.6.6.2

Add $x$ and $x$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - x (x - 3) (x - 6) \frac{d}{d x} | x (x - 6) |}{{| x (x - 6) |}^{2}})$

Step 3.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x (x - 6)$ .

Tap for more steps...

Step 3.7.1

To apply the Chain Rule, set $u$ as $x (x - 6)$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - x (x - 3) (x - 6) (\frac{d}{d u} (| u |) \frac{d}{d x} (x (x - 6)))}{{| x (x - 6) |}^{2}})$

Step 3.7.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - x (x - 3) (x - 6) (\frac{u}{| u |} \cdot \frac{d}{d x} (x (x - 6)))}{{| x (x - 6) |}^{2}})$

Step 3.7.3

Replace all occurrences of $u$ with $x (x - 6)$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - x (x - 3) (x - 6) (\frac{x (x - 6)}{| x (x - 6) |} \cdot \frac{d}{d x} (x (x - 6)))}{{| x (x - 6) |}^{2}})$

Step 3.8

Combine $\frac{x (x - 6)}{| x (x - 6) |}$ and $x$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x (x - 6) x}{| x (x - 6) |} \cdot ((x - 3) (x - 6)) \frac{d}{d x} (x (x - 6))}{{| x (x - 6) |}^{2}})$

Step 3.9

Raise $x$ to the power of $1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x \cdot x (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6)) \frac{d}{d x} (x (x - 6))}{{| x (x - 6) |}^{2}})$

Step 3.10

Raise $x$ to the power of $1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x \cdot x (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6)) \frac{d}{d x} (x (x - 6))}{{| x (x - 6) |}^{2}})$

Step 3.11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{1 + 1} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6)) \frac{d}{d x} (x (x - 6))}{{| x (x - 6) |}^{2}})$

Step 3.12

Add $1$ and $1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6)) \frac{d}{d x} (x (x - 6))}{{| x (x - 6) |}^{2}})$

Step 3.13

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x - 6$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (x \frac{d}{d x} (x - 6) + (x - 6) \frac{d}{d x} (x)))}{{| x (x - 6) |}^{2}})$

Step 3.14

Differentiate.

Tap for more steps...

Step 3.14.1

By the Sum Rule, the derivative of $x - 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 6]$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (x (\frac{d}{d x} (x) + \frac{d}{d x} (- 6)) + (x - 6) \frac{d}{d x} (x)))}{{| x (x - 6) |}^{2}})$

Step 3.14.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (x (1 + \frac{d}{d x} (- 6)) + (x - 6) \frac{d}{d x} (x)))}{{| x (x - 6) |}^{2}})$

Step 3.14.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (x (1 + 0) + (x - 6) \frac{d}{d x} (x)))}{{| x (x - 6) |}^{2}})$

Step 3.14.4

Simplify the expression.

Tap for more steps...

Step 3.14.4.1

Add $1$ and $0$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (x \cdot 1 + (x - 6) \frac{d}{d x} (x)))}{{| x (x - 6) |}^{2}})$

Step 3.14.4.2

Multiply $x$ by $1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (x + (x - 6) \frac{d}{d x} (x)))}{{| x (x - 6) |}^{2}})$

Step 3.14.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (x + (x - 6) \cdot 1))}{{| x (x - 6) |}^{2}})$

Step 3.14.6

Simplify terms.

Tap for more steps...

Step 3.14.6.1

Multiply $x - 6$ by $1$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (x + x - 6))}{{| x (x - 6) |}^{2}})$

Step 3.14.6.2

Add $x$ and $x$ .

$f'' (x) = 2 (\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6))}{{| x (x - 6) |}^{2}})$

Step 3.14.6.3

Combine $2$ and $\frac{| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} (x - 3) (x - 6) (2 x - 6)}{{| x (x - 6) |}^{2}}$ .

$f'' (x) = \frac{2 (| x (x - 6) | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x (x - 6) |}^{2}}$

Step 3.15

Simplify.

Tap for more steps...

Step 3.15.1

Apply the distributive property.

$f'' (x) = \frac{2 (| x \cdot x + x \cdot - 6 | (x (x - 3) + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x (x - 6) |}^{2}}$

Step 3.15.2

Apply the distributive property.

$f'' (x) = \frac{2 (| x \cdot x + x \cdot - 6 | (x \cdot x + x \cdot - 3 + (x - 6) (2 x - 3)) - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x (x - 6) |}^{2}}$

Step 3.15.3

Apply the distributive property.

$f'' (x) = \frac{2 (| x \cdot x + x \cdot - 6 | (x \cdot x + x \cdot - 3 + (x - 6) (2 x - 3)) - \frac{x^{2} x + x^{2} \cdot - 6}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x (x - 6) |}^{2}}$

Step 3.15.4

Apply the distributive property.

$f'' (x) = \frac{2 (| x \cdot x + x \cdot - 6 | (x \cdot x + x \cdot - 3 + (x - 6) (2 x - 3)) - \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x (x - 6) |}^{2}}$

Step 3.15.5

Apply the distributive property.

$f'' (x) = \frac{2 (| x \cdot x + x \cdot - 6 | (x \cdot x + x \cdot - 3 + (x - 6) (2 x - 3))) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x (x - 6) |}^{2}}$

Step 3.15.6

Apply the distributive property.

$f'' (x) = \frac{2 (| x \cdot x + x \cdot - 6 | (x \cdot x + x \cdot - 3 + (x - 6) (2 x - 3))) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7

Simplify the numerator.

Tap for more steps...

Step 3.15.7.1

Simplify each term.

Tap for more steps...

Step 3.15.7.1.1

Simplify each term.

Tap for more steps...

Step 3.15.7.1.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{2 (| x^{2} + x \cdot - 6 | (x \cdot x + x \cdot - 3 + (x - 6) (2 x - 3))) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.1.2

Move $- 6$ to the left of $x$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x \cdot x + x \cdot - 3 + (x - 6) (2 x - 3))) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2

Simplify each term.

Tap for more steps...

Step 3.15.7.1.2.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} + x \cdot - 3 + (x - 6) (2 x - 3))) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.2

Move $- 3$ to the left of $x$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 \cdot x + (x - 6) (2 x - 3))) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.3

Expand $(x - 6) (2 x - 3)$ using the FOIL Method.

Tap for more steps...

Step 3.15.7.1.2.3.1

Apply the distributive property.

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + x (2 x - 3) - 6 (2 x - 3))) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.3.2

Apply the distributive property.

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + x (2 x) + x \cdot - 3 - 6 (2 x - 3))) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.3.3

Apply the distributive property.

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + x (2 x) + x \cdot - 3 - 6 (2 x) - 6 \cdot - 3)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.4

Simplify and combine like terms.

Tap for more steps...

Step 3.15.7.1.2.4.1

Simplify each term.

Tap for more steps...

Step 3.15.7.1.2.4.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + 2 x \cdot x + x \cdot - 3 - 6 (2 x) - 6 \cdot - 3)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.4.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.1.2.4.1.2.1

Move $x$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + 2 (x \cdot x) + x \cdot - 3 - 6 (2 x) - 6 \cdot - 3)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.4.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + 2 x^{2} + x \cdot - 3 - 6 (2 x) - 6 \cdot - 3)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.4.1.3

Move $- 3$ to the left of $x$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + 2 x^{2} - 3 \cdot x - 6 (2 x) - 6 \cdot - 3)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.4.1.4

Multiply $2$ by $- 6$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + 2 x^{2} - 3 x - 12 x - 6 \cdot - 3)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.4.1.5

Multiply $- 6$ by $- 3$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + 2 x^{2} - 3 x - 12 x + 18)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.2.4.2

Subtract $12 x$ from $- 3 x$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (x^{2} - 3 x + 2 x^{2} - 15 x + 18)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.3

Add $x^{2}$ and $2 x^{2}$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (3 x^{2} - 3 x - 15 x + 18)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.4

Subtract $15 x$ from $- 3 x$ .

$f'' (x) = \frac{2 (| x^{2} - 6 x | (3 x^{2} - 18 x + 18)) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.5

Apply the distributive property.

$f'' (x) = \frac{2 (| x^{2} - 6 x | (3 x^{2}) + | x^{2} - 6 x | (- 18 x) + | x^{2} - 6 x | \cdot 18) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.6

Simplify.

Tap for more steps...

Step 3.15.7.1.6.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 (3 | x^{2} - 6 x | x^{2} + | x^{2} - 6 x | (- 18 x) + | x^{2} - 6 x | \cdot 18) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.6.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 (3 | x^{2} - 6 x | x^{2} - 18 | x^{2} - 6 x | x + | x^{2} - 6 x | \cdot 18) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.6.3

Move $18$ to the left of $| x^{2} - 6 x |$ .

$f'' (x) = \frac{2 (3 | x^{2} - 6 x | x^{2} - 18 | x^{2} - 6 x | x + 18 \cdot | x^{2} - 6 x |) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.7

Apply the distributive property.

$f'' (x) = \frac{2 (3 | x^{2} - 6 x | x^{2}) + 2 (- 18 | x^{2} - 6 x | x) + 2 (18 | x^{2} - 6 x |) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.8

Simplify.

Tap for more steps...

Step 3.15.7.1.8.1

Multiply $3$ by $2$ .

$f'' (x) = \frac{6 (| x^{2} - 6 x | x^{2}) + 2 (- 18 | x^{2} - 6 x | x) + 2 (18 | x^{2} - 6 x |) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.8.2

Multiply $- 18$ by $2$ .

$f'' (x) = \frac{6 (| x^{2} - 6 x | x^{2}) - 36 (| x^{2} - 6 x | x) + 2 (18 | x^{2} - 6 x |) + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.8.3

Multiply $18$ by $2$ .

$f'' (x) = \frac{6 (| x^{2} - 6 x | x^{2}) - 36 (| x^{2} - 6 x | x) + 36 | x^{2} - 6 x | + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.9

Remove parentheses.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{2} x + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10

Simplify the numerator.

Tap for more steps...

Step 3.15.7.1.10.1

Factor $x^{3}$ out of $x^{2} x + x^{2} \cdot - 6$ .

Tap for more steps...

Step 3.15.7.1.10.1.1

Factor $x^{3}$ out of $x^{2} x$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (x^{- 1} x) + x^{2} \cdot - 6}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.1.2

Factor $x^{3}$ out of $x^{2} \cdot - 6$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (x^{- 1} x) + x^{3} (x^{- 1} \cdot - 6)}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.1.3

Factor $x^{3}$ out of $x^{3} (x^{- 1} x) + x^{3} (x^{- 1} \cdot - 6)$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (x^{- 1} x + x^{- 1} \cdot - 6)}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.2

Multiply $x^{- 1}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.1.10.2.1

Multiply $x^{- 1}$ by $x$ .

Tap for more steps...

Step 3.15.7.1.10.2.1.1

Raise $x$ to the power of $1$ .

Step 3.15.7.1.10.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (x^{- 1 + 1} + x^{- 1} \cdot - 6)}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.2.2

Add $- 1$ and $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (x^{0} + x^{- 1} \cdot - 6)}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.3

Simplify $x^{0}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (1 + x^{- 1} \cdot - 6)}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (1 + \frac{1}{x} \cdot - 6)}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.5

Combine $\frac{1}{x}$ and $- 6$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (1 + \frac{- 6}{x})}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.6

Move the negative in front of the fraction.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (1 - \frac{6}{x})}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.7

Write $1$ as a fraction with a common denominator.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (\frac{x}{x} - \frac{6}{x})}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.10.8

Combine the numerators over the common denominator.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (\frac{x - 6}{x})}{| x \cdot x + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.11

Simplify the denominator.

Tap for more steps...

Step 3.15.7.1.11.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (\frac{x - 6}{x})}{| x^{2} + x \cdot - 6 |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.11.2

Move $- 6$ to the left of $x$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (\frac{x - 6}{x})}{| x^{2} - 6 \cdot x |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.11.3

Factor $x$ out of $x^{2} - 6 x$ .

Tap for more steps...

Step 3.15.7.1.11.3.1

Factor $x$ out of $x^{2}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (\frac{x - 6}{x})}{| x \cdot x - 6 x |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.11.3.2

Factor $x$ out of $- 6 x$ .

Step 3.15.7.1.11.3.3

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{3} (\frac{x - 6}{x})}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.11.4

Apply the distributive property.

Step 3.15.7.1.11.5

Multiply $x$ by $x$ .

Step 3.15.7.1.11.6

Move $- 6$ to the left of $x$ .

Step 3.15.7.1.11.7

Factor $x$ out of $x^{2} - 6 x$ .

Tap for more steps...

Step 3.15.7.1.11.7.1

Factor $x$ out of $x^{2}$ .

Step 3.15.7.1.11.7.2

Factor $x$ out of $- 6 x$ .

Step 3.15.7.1.11.7.3

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

Step 3.15.7.1.12

Combine $x^{3}$ and $\frac{x - 6}{x}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{\frac{x^{3} (x - 6)}{x}}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.13

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 3.15.7.1.13.1

Reduce the expression $\frac{x^{3} (x - 6)}{x}$ by cancelling the common factors.

Tap for more steps...

Step 3.15.7.1.13.1.1

Factor $x$ out of $x^{3} (x - 6)$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{\frac{x (x^{2} (x - 6))}{x}}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.13.1.2

Raise $x$ to the power of $1$ .

Step 3.15.7.1.13.1.3

Factor $x$ out of $x^{1}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{\frac{x (x^{2} (x - 6))}{x \cdot 1}}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.13.1.4

Cancel the common factor.

Step 3.15.7.1.13.1.5

Rewrite the expression.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{\frac{x^{2} (x - 6)}{1}}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.13.2

Divide $x^{2} (x - 6)$ by $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot ((x - 3) (x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.14

Apply the distributive property.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 ((- \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot x - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot - 3) (x - 6) (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.15

Multiply $- \frac{x^{2} (x - 6)}{| x (x - 6) |} x$ .

Tap for more steps...

Step 3.15.7.1.15.1

Combine $x$ and $\frac{x^{2} (x - 6)}{| x (x - 6) |}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 ((- \frac{x (x^{2} (x - 6))}{| x (x - 6) |} - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot - 3) (x - 6) (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.15.2

Raise $x$ to the power of $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 ((- \frac{x^{2} x (x - 6)}{| x (x - 6) |} - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot - 3) (x - 6) (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.15.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 ((- \frac{x^{2 + 1} (x - 6)}{| x (x - 6) |} - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot - 3) (x - 6) (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.15.4

Add $2$ and $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 ((- \frac{x^{3} (x - 6)}{| x (x - 6) |} - \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot - 3) (x - 6) (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.16

Multiply $- \frac{x^{2} (x - 6)}{| x (x - 6) |} \cdot - 3$ .

Tap for more steps...

Step 3.15.7.1.16.1

Multiply $- 3$ by $- 1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 ((- \frac{x^{3} (x - 6)}{| x (x - 6) |} + 3 (\frac{x^{2} (x - 6)}{| x (x - 6) |})) (x - 6) (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.16.2

Combine $3$ and $\frac{x^{2} (x - 6)}{| x (x - 6) |}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 ((- \frac{x^{3} (x - 6)}{| x (x - 6) |} + \frac{3 (x^{2} (x - 6))}{| x (x - 6) |}) (x - 6) (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.17

Combine the numerators over the common denominator.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{3} (x - 6) + 3 (x^{2} (x - 6))}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18

Simplify each term.

Tap for more steps...

Step 3.15.7.1.18.1

Apply the distributive property.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{3} x - x^{3} \cdot - 6 + 3 (x^{2} (x - 6))}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.2

Multiply $x^{3}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.1.18.2.1

Move $x$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- (x \cdot x^{3}) - x^{3} \cdot - 6 + 3 (x^{2} (x - 6))}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.2.2

Multiply $x$ by $x^{3}$ .

Tap for more steps...

Step 3.15.7.1.18.2.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.1.18.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{1 + 3} - x^{3} \cdot - 6 + 3 (x^{2} (x - 6))}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.2.3

Add $1$ and $3$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{4} - x^{3} \cdot - 6 + 3 (x^{2} (x - 6))}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.3

Multiply $- 6$ by $- 1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{4} + 6 x^{3} + 3 (x^{2} (x - 6))}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.4

Apply the distributive property.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{4} + 6 x^{3} + 3 (x^{2} x + x^{2} \cdot - 6)}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.5

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.1.18.5.1

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 3.15.7.1.18.5.1.1

Raise $x$ to the power of $1$ .

Step 3.15.7.1.18.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{4} + 6 x^{3} + 3 (x^{2 + 1} + x^{2} \cdot - 6)}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.5.2

Add $2$ and $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{4} + 6 x^{3} + 3 (x^{3} + x^{2} \cdot - 6)}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.6

Move $- 6$ to the left of $x^{2}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{4} + 6 x^{3} + 3 (x^{3} - 6 \cdot x^{2})}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.7

Apply the distributive property.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{4} + 6 x^{3} + 3 x^{3} + 3 (- 6 x^{2})}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.18.8

Multiply $- 6$ by $3$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{4} + 6 x^{3} + 3 x^{3} - 18 x^{2}}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.19

Add $6 x^{3}$ and $3 x^{3}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- x^{4} + 9 x^{3} - 18 x^{2}}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20

Simplify the numerator.

Tap for more steps...

Step 3.15.7.1.20.1

Factor $x^{2}$ out of $- x^{4} + 9 x^{3} - 18 x^{2}$ .

Tap for more steps...

Step 3.15.7.1.20.1.1

Factor $x^{2}$ out of $- x^{4}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x^{2}) + 9 x^{3} - 18 x^{2}}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.1.2

Factor $x^{2}$ out of $9 x^{3}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x^{2}) + x^{2} (9 x) - 18 x^{2}}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.1.3

Factor $x^{2}$ out of $- 18 x^{2}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x^{2}) + x^{2} (9 x) + x^{2} \cdot - 18}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.1.4

Factor $x^{2}$ out of $x^{2} (- x^{2}) + x^{2} (9 x)$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x^{2} + 9 x) + x^{2} \cdot - 18}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.1.5

Factor $x^{2}$ out of $x^{2} (- x^{2} + 9 x) + x^{2} \cdot - 18$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x^{2} + 9 x - 18)}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.2

Factor by grouping.

Tap for more steps...

Step 3.15.7.1.20.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 18 = 18$ and whose sum is $b = 9$ .

Tap for more steps...

Step 3.15.7.1.20.2.1.1

Factor $9$ out of $9 x$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x^{2} + 9 (x) - 18)}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.2.1.2

Rewrite $9$ as $3$ plus $6$

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x^{2} + (3 + 6) x - 18)}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.2.1.3

Apply the distributive property.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x^{2} + 3 x + 6 x - 18)}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.15.7.1.20.2.2.1

Group the first two terms and the last two terms.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} ((- x^{2} + 3 x) + 6 x - 18)}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.2.2.2

Factor out the greatest common factor (GCF) from each group.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (x (- x + 3) - 6 (- x + 3))}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.20.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 3$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} ((- x + 3) (x - 6))}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) (x - 6)}{| x (x - 6) |} \cdot ((x - 6) (2 x - 6)))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.21

Multiply $\frac{x^{2} (- x + 3) (x - 6)}{| x (x - 6) |}$ by $x - 6$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) (x - 6) (x - 6)}{| x (x - 6) |} \cdot (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.22

Simplify the numerator.

Tap for more steps...

Step 3.15.7.1.22.1

Raise $x - 6$ to the power of $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) ((x - 6) (x - 6))}{| x (x - 6) |} \cdot (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.22.2

Raise $x - 6$ to the power of $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) ((x - 6) (x - 6))}{| x (x - 6) |} \cdot (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.22.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) {(x - 6)}^{1 + 1}}{| x (x - 6) |} \cdot (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.22.4

Add $1$ and $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) {(x - 6)}^{2}}{| x (x - 6) |} \cdot (2 x - 6))}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.23

Multiply $\frac{x^{2} (- x + 3) {(x - 6)}^{2}}{| x (x - 6) |}$ by $2 x - 6$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) {(x - 6)}^{2} (2 x - 6)}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24

Simplify the numerator.

Tap for more steps...

Step 3.15.7.1.24.1

Factor $2$ out of $2 x - 6$ .

Tap for more steps...

Step 3.15.7.1.24.1.1

Factor $2$ out of $2 x$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) {(x - 6)}^{2} (2 (x) - 6)}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.1.2

Factor $2$ out of $- 6$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) {(x - 6)}^{2} (2 x + 2 \cdot - 3)}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.1.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) {(x - 6)}^{2} (2 (x - 3))}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- x + 3) {(x - 6)}^{2} \cdot (2 (x - 3))}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.2

Combine exponents.

Tap for more steps...

Step 3.15.7.1.24.2.1

Factor $- 1$ out of $- x$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- (x) + 3) {(x - 6)}^{2} \cdot (2 (x - 3))}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.2.2

Rewrite $3$ as $- 1 (- 3)$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- (x) - 1 \cdot - 3) {(x - 6)}^{2} \cdot (2 (x - 3))}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.2.3

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- (x - 3)) {(x - 6)}^{2} \cdot (2 (x - 3))}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.2.4

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} (- 1 (x - 3)) {(x - 6)}^{2} \cdot (2 (x - 3))}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.2.5

Raise $x - 3$ to the power of $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} \cdot (- 1 {(x - 6)}^{2}) \cdot (2 ((x - 3) (x - 3)))}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.2.6

Raise $x - 3$ to the power of $1$ .

Step 3.15.7.1.24.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} \cdot (- 1 {(x - 6)}^{2}) \cdot (2 {(x - 3)}^{1 + 1})}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.2.8

Add $1$ and $1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} \cdot (- 1 {(x - 6)}^{2}) \cdot (2 {(x - 3)}^{2})}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.24.2.9

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{x^{2} \cdot (- 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.25

Move $- 2$ to the left of $x^{2}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (\frac{- 2 \cdot (x^{2} {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.26

Move the negative in front of the fraction.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + 2 (- \frac{(2) \cdot (x^{2} {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |})}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.27

Multiply $2 (- \frac{2 x^{2} {(x - 6)}^{2} {(x - 3)}^{2}}{| x (x - 6) |})$ .

Tap for more steps...

Step 3.15.7.1.27.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | - 2 \frac{2 x^{2} {(x - 6)}^{2} {(x - 3)}^{2}}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.27.2

Combine $- 2$ and $\frac{2 x^{2} {(x - 6)}^{2} {(x - 3)}^{2}}{| x (x - 6) |}$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{- 2 (2 x^{2} {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.27.3

Multiply $2$ by $- 2$ .

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{- 4 (x^{2} {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.1.28

Move the negative in front of the fraction.

$f'' (x) = \frac{6 | x^{2} - 6 x | x^{2} - 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | - \frac{4 x^{2} {(x - 6)}^{2} {(x - 3)}^{2}}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.2

To write $6 | x^{2} - 6 x | x^{2}$ as a fraction with a common denominator, multiply by $\frac{| x (x - 6) |}{| x (x - 6) |}$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{6 | x^{2} - 6 x | x^{2} | x (x - 6) |}{| x (x - 6) |} - \frac{4 x^{2} {(x - 6)}^{2} {(x - 3)}^{2}}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.3

Combine the numerators over the common denominator.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{6 | x^{2} - 6 x | x^{2} | x (x - 6) | - 4 x^{2} {(x - 6)}^{2} {(x - 3)}^{2}}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4

Simplify the numerator.

Tap for more steps...

Step 3.15.7.4.1

Factor $2 x^{2}$ out of $6 | x^{2} - 6 x | x^{2} | x (x - 6) | - 4 x^{2} {(x - 6)}^{2} {(x - 3)}^{2}$ .

Tap for more steps...

Step 3.15.7.4.1.1

Factor $2 x^{2}$ out of $6 | x^{2} - 6 x | x^{2} | x (x - 6) |$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{2} - 6 x | | x (x - 6) |) - 4 x^{2} {(x - 6)}^{2} {(x - 3)}^{2}}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.1.2

Factor $2 x^{2}$ out of $- 4 x^{2} {(x - 6)}^{2} {(x - 3)}^{2}$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{2} - 6 x | | x (x - 6) |) + 2 x^{2} (- 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.1.3

Factor $2 x^{2}$ out of $2 x^{2} (3 | x^{2} - 6 x | | x (x - 6) |) + 2 x^{2} (- 2 {(x - 6)}^{2} {(x - 3)}^{2})$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{2} - 6 x | | x (x - 6) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.2

To multiply absolute values, multiply the terms inside each absolute value.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x (x - 6) (x^{2} - 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3

Simplify each term.

Tap for more steps...

Step 3.15.7.4.3.1

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | (x \cdot x + x \cdot - 6) (x^{2} - 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | (x^{2} + x \cdot - 6) (x^{2} - 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.3

Move $- 6$ to the left of $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | (x^{2} - 6 \cdot x) (x^{2} - 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.4

Expand $(x^{2} - 6 x) (x^{2} - 6 x)$ using the FOIL Method.

Tap for more steps...

Step 3.15.7.4.3.4.1

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{2} (x^{2} - 6 x) - 6 x (x^{2} - 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.4.2

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{2} x^{2} + x^{2} (- 6 x) - 6 x (x^{2} - 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.4.3

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{2} x^{2} + x^{2} (- 6 x) - 6 x \cdot x^{2} - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5

Simplify and combine like terms.

Tap for more steps...

Step 3.15.7.4.3.5.1

Simplify each term.

Tap for more steps...

Step 3.15.7.4.3.5.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.4.3.5.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{2 + 2} + x^{2} (- 6 x) - 6 x \cdot x^{2} - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} + x^{2} (- 6 x) - 6 x \cdot x^{2} - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{2} x - 6 x \cdot x^{2} - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.4.3.5.1.3.1

Move $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 (x \cdot x^{2}) - 6 x \cdot x^{2} - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.3.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 3.15.7.4.3.5.1.3.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.4.3.5.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{1 + 2} - 6 x \cdot x^{2} - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{3} - 6 x \cdot x^{2} - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.4

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.4.3.5.1.4.1

Move $x^{2}$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{3} - 6 (x^{2} x) - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.4.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 3.15.7.4.3.5.1.4.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.4.3.5.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{3} - 6 x^{2 + 1} - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.4.3

Add $2$ and $1$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{3} - 6 x^{3} - 6 x (- 6 x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{3} - 6 x^{3} - 6 \cdot (- 6 x \cdot x) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.6

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.4.3.5.1.6.1

Move $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{3} - 6 x^{3} - 6 \cdot (- 6 (x \cdot x)) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.6.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{3} - 6 x^{3} - 6 \cdot (- 6 x^{2}) | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.1.7

Multiply $- 6$ by $- 6$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 6 x^{3} - 6 x^{3} + 36 x^{2} | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.5.2

Subtract $6 x^{3}$ from $- 6 x^{3}$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 {(x - 6)}^{2} {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.6

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 ((x - 6) (x - 6)) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.7

Expand $(x - 6) (x - 6)$ using the FOIL Method.

Tap for more steps...

Step 3.15.7.4.3.7.1

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x (x - 6) - 6 (x - 6)) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.7.2

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x \cdot x + x \cdot - 6 - 6 (x - 6)) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.7.3

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.8

Simplify and combine like terms.

Tap for more steps...

Step 3.15.7.4.3.8.1

Simplify each term.

Tap for more steps...

Step 3.15.7.4.3.8.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.8.1.2

Move $- 6$ to the left of $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.8.1.3

Multiply $- 6$ by $- 6$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x^{2} - 6 x - 6 x + 36) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.8.2

Subtract $6 x$ from $- 6 x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x^{2} - 12 x + 36) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.9

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} - 2 (- 12 x) - 2 \cdot 36) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.10

Simplify.

Tap for more steps...

Step 3.15.7.4.3.10.1

Multiply $- 12$ by $- 2$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 2 \cdot 36) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.10.2

Multiply $- 2$ by $36$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 72) {(x - 3)}^{2})}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.11

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 72) ((x - 3) (x - 3)))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.12

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Tap for more steps...

Step 3.15.7.4.3.12.1

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 72) (x (x - 3) - 3 (x - 3)))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.12.2

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 72) (x \cdot x + x \cdot - 3 - 3 (x - 3)))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.12.3

Apply the distributive property.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 72) (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.13

Simplify and combine like terms.

Tap for more steps...

Step 3.15.7.4.3.13.1

Simplify each term.

Tap for more steps...

Step 3.15.7.4.3.13.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 72) (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.13.1.2

Move $- 3$ to the left of $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 72) (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.13.1.3

Multiply $- 3$ by $- 3$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 72) (x^{2} - 3 x - 3 x + 9))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.13.2

Subtract $3 x$ from $- 3 x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | + (- 2 x^{2} + 24 x - 72) (x^{2} - 6 x + 9))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.14

Expand $(- 2 x^{2} + 24 x - 72) (x^{2} - 6 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{2} x^{2} - 2 x^{2} (- 6 x) - 2 x^{2} \cdot 9 + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15

Simplify each term.

Tap for more steps...

Step 3.15.7.4.3.15.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.4.3.15.1.1

Move $x^{2}$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x^{2} x^{2}) - 2 x^{2} (- 6 x) - 2 x^{2} \cdot 9 + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{2 + 2} - 2 x^{2} (- 6 x) - 2 x^{2} \cdot 9 + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.1.3

Add $2$ and $2$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} - 2 x^{2} (- 6 x) - 2 x^{2} \cdot 9 + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} - 2 \cdot (- 6 x^{2} x) - 2 x^{2} \cdot 9 + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.3

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.4.3.15.3.1

Move $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} - 2 \cdot (- 6 (x \cdot x^{2})) - 2 x^{2} \cdot 9 + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.3.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 3.15.7.4.3.15.3.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.4.3.15.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} - 2 \cdot (- 6 x^{1 + 2}) - 2 x^{2} \cdot 9 + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} - 2 \cdot (- 6 x^{3}) - 2 x^{2} \cdot 9 + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.4

Multiply $- 2$ by $- 6$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 2 x^{2} \cdot 9 + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.5

Multiply $9$ by $- 2$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x \cdot x^{2} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.6

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.4.3.15.6.1

Move $x^{2}$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 (x^{2} x) + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.6.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 3.15.7.4.3.15.6.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.4.3.15.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x^{2 + 1} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.6.3

Add $2$ and $1$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x^{3} + 24 x (- 6 x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.7

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x^{3} + 24 \cdot (- 6 x \cdot x) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.8

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.4.3.15.8.1

Move $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x^{3} + 24 \cdot (- 6 (x \cdot x)) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.8.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x^{3} + 24 \cdot (- 6 x^{2}) + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.9

Multiply $24$ by $- 6$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x^{3} - 144 x^{2} + 24 x \cdot 9 - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.10

Multiply $9$ by $24$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x^{3} - 144 x^{2} + 216 x - 72 x^{2} - 72 (- 6 x) - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.11

Multiply $- 6$ by $- 72$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x^{3} - 144 x^{2} + 216 x - 72 x^{2} + 432 x - 72 \cdot 9)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.15.12

Multiply $- 72$ by $9$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 12 x^{3} - 18 x^{2} + 24 x^{3} - 144 x^{2} + 216 x - 72 x^{2} + 432 x - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.16

Add $12 x^{3}$ and $24 x^{3}$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 36 x^{3} - 18 x^{2} - 144 x^{2} + 216 x - 72 x^{2} + 432 x - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.17

Subtract $144 x^{2}$ from $- 18 x^{2}$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 36 x^{3} - 162 x^{2} + 216 x - 72 x^{2} + 432 x - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.18

Subtract $72 x^{2}$ from $- 162 x^{2}$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 36 x^{3} - 234 x^{2} + 216 x + 432 x - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.3.19

Add $216 x$ and $432 x$ .

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (3 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.4.4

Reorder terms.

$f'' (x) = \frac{- 36 | x^{2} - 6 x | x + 36 | x^{2} - 6 x | + \frac{2 x^{2} (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.5

To write $- 36 | x^{2} - 6 x | x$ as a fraction with a common denominator, multiply by $\frac{| x (x - 6) |}{| x (x - 6) |}$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | - 36 | x^{2} - 6 x | x \cdot \frac{| x (x - 6) |}{| x (x - 6) |} + \frac{2 x^{2} (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.6

Combine $- 36 | x^{2} - 6 x | x$ and $\frac{| x (x - 6) |}{| x (x - 6) |}$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{- 36 | x^{2} - 6 x | x | x (x - 6) |}{| x (x - 6) |} + \frac{2 x^{2} (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.7

Combine the numerators over the common denominator.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{- 36 | x^{2} - 6 x | x | x (x - 6) | + 2 x^{2} (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8

Simplify the numerator.

Tap for more steps...

Step 3.15.7.8.1

Factor $2 x$ out of $- 36 | x^{2} - 6 x | x | x (x - 6) | + 2 x^{2} (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648)$ .

Tap for more steps...

Step 3.15.7.8.1.1

Factor $2 x$ out of $- 36 | x^{2} - 6 x | x | x (x - 6) |$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{2} - 6 x | | x (x - 6) |) + 2 x^{2} (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.1.2

Factor $2 x$ out of $2 x^{2} (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648)$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{2} - 6 x | | x (x - 6) |) + 2 x (x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.1.3

Factor $2 x$ out of $2 x (- 18 | x^{2} - 6 x | | x (x - 6) |) + 2 x (x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{2} - 6 x | | x (x - 6) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.2

To multiply absolute values, multiply the terms inside each absolute value.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x (x - 6) (x^{2} - 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3

Simplify each term.

Tap for more steps...

Step 3.15.7.8.3.1

Apply the distributive property.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | (x \cdot x + x \cdot - 6) (x^{2} - 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | (x^{2} + x \cdot - 6) (x^{2} - 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.3

Move $- 6$ to the left of $x$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | (x^{2} - 6 \cdot x) (x^{2} - 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.4

Expand $(x^{2} - 6 x) (x^{2} - 6 x)$ using the FOIL Method.

Tap for more steps...

Step 3.15.7.8.3.4.1

Apply the distributive property.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{2} (x^{2} - 6 x) - 6 x (x^{2} - 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.4.2

Apply the distributive property.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{2} x^{2} + x^{2} (- 6 x) - 6 x (x^{2} - 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.4.3

Apply the distributive property.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{2} x^{2} + x^{2} (- 6 x) - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5

Simplify and combine like terms.

Tap for more steps...

Step 3.15.7.8.3.5.1

Simplify each term.

Tap for more steps...

Step 3.15.7.8.3.5.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.8.3.5.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{2 + 2} + x^{2} (- 6 x) - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} + x^{2} (- 6 x) - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{2} x - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.8.3.5.1.3.1

Move $x$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 (x \cdot x^{2}) - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.3.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 3.15.7.8.3.5.1.3.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.8.3.5.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{1 + 2} - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{3} - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.4

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.8.3.5.1.4.1

Move $x^{2}$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{3} - 6 (x^{2} x) - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.4.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 3.15.7.8.3.5.1.4.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.8.3.5.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{3} - 6 x^{2 + 1} - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.4.3

Add $2$ and $1$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{3} - 6 x^{3} - 6 x (- 6 x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{3} - 6 x^{3} - 6 \cdot (- 6 x \cdot x) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.6

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.8.3.5.1.6.1

Move $x$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{3} - 6 x^{3} - 6 \cdot (- 6 (x \cdot x)) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.6.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{3} - 6 x^{3} - 6 \cdot (- 6 x^{2}) | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.1.7

Multiply $- 6$ by $- 6$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 6 x^{3} - 6 x^{3} + 36 x^{2} | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.5.2

Subtract $6 x^{3}$ from $- 6 x^{3}$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | + x (- 2 x^{4} + 36 x^{3} - 234 x^{2} + 648 x + 3 | x^{4} - 12 x^{3} + 36 x^{2} | - 648))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.6

Apply the distributive property.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | + x (- 2 x^{4}) + x (36 x^{3}) + x (- 234 x^{2}) + x (648 x) + x (3 | x^{4} - 12 x^{3} + 36 x^{2} |) + x \cdot - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.7

Simplify.

Tap for more steps...

Step 3.15.7.8.3.7.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{4} + x (36 x^{3}) + x (- 234 x^{2}) + x (648 x) + x (3 | x^{4} - 12 x^{3} + 36 x^{2} |) + x \cdot - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.7.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{4} + 36 x \cdot x^{3} + x (- 234 x^{2}) + x (648 x) + x (3 | x^{4} - 12 x^{3} + 36 x^{2} |) + x \cdot - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.7.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{4} + 36 x \cdot x^{3} - 234 x \cdot x^{2} + x (648 x) + x (3 | x^{4} - 12 x^{3} + 36 x^{2} |) + x \cdot - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.7.4

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{4} + 36 x \cdot x^{3} - 234 x \cdot x^{2} + 648 x \cdot x + x (3 | x^{4} - 12 x^{3} + 36 x^{2} |) + x \cdot - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.7.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{4} + 36 x \cdot x^{3} - 234 x \cdot x^{2} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | + x \cdot - 648)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.7.6

Move $- 648$ to the left of $x$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{4} + 36 x \cdot x^{3} - 234 x \cdot x^{2} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8

Simplify each term.

Tap for more steps...

Step 3.15.7.8.3.8.1

Multiply $x$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.8.3.8.1.1

Move $x^{4}$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x^{4} x) + 36 x \cdot x^{3} - 234 x \cdot x^{2} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.1.2

Multiply $x^{4}$ by $x$ .

Tap for more steps...

Step 3.15.7.8.3.8.1.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.8.3.8.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{4 + 1} + 36 x \cdot x^{3} - 234 x \cdot x^{2} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.1.3

Add $4$ and $1$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 x \cdot x^{3} - 234 x \cdot x^{2} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.2

Multiply $x$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.8.3.8.2.1

Move $x^{3}$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 (x^{3} x) - 234 x \cdot x^{2} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.2.2

Multiply $x^{3}$ by $x$ .

Tap for more steps...

Step 3.15.7.8.3.8.2.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.8.3.8.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 x^{3 + 1} - 234 x \cdot x^{2} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.2.3

Add $3$ and $1$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 x^{4} - 234 x \cdot x^{2} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.3

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.8.3.8.3.1

Move $x^{2}$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 x^{4} - 234 (x^{2} x) + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.3.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 3.15.7.8.3.8.3.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.8.3.8.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 x^{4} - 234 x^{2 + 1} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.3.3

Add $2$ and $1$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x \cdot x + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.4

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.8.3.8.4.1

Move $x$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 (x \cdot x) + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.3.8.4.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 \cdot x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.8.4

Reorder terms.

$f'' (x) = \frac{36 | x^{2} - 6 x | + \frac{2 x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.9

To write $36 | x^{2} - 6 x |$ as a fraction with a common denominator, multiply by $\frac{| x (x - 6) |}{| x (x - 6) |}$ .

$f'' (x) = \frac{\frac{36 | x^{2} - 6 x | | x (x - 6) |}{| x (x - 6) |} + \frac{2 x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.10

Combine the numerators over the common denominator.

$f'' (x) = \frac{\frac{36 | x^{2} - 6 x | | x (x - 6) | + 2 x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11

Simplify the numerator.

Tap for more steps...

Step 3.15.7.11.1

Factor $2$ out of $36 | x^{2} - 6 x | | x (x - 6) | + 2 x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)$ .

Tap for more steps...

Step 3.15.7.11.1.1

Factor $2$ out of $36 | x^{2} - 6 x | | x (x - 6) |$ .

$f'' (x) = \frac{\frac{2 (18 | x^{2} - 6 x | | x (x - 6) |) + 2 x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.1.2

Factor $2$ out of $2 x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)$ .

$f'' (x) = \frac{\frac{2 (18 | x^{2} - 6 x | | x (x - 6) |) + 2 (x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.1.3

Factor $2$ out of $2 (18 | x^{2} - 6 x | | x (x - 6) |) + 2 (x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))$ .

$f'' (x) = \frac{\frac{2 (18 | x^{2} - 6 x | | x (x - 6) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.2

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (18 | x^{2} - 6 x | | x \cdot x + x \cdot - 6 | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.3

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 (18 | x^{2} - 6 x | | x^{2} + x \cdot - 6 | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.4

Move $- 6$ to the left of $x$ .

$f'' (x) = \frac{\frac{2 (18 | x^{2} - 6 x | | x^{2} - 6 \cdot x | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.5

Multiply $18 | x^{2} - 6 x | | x^{2} - 6 x |$ .

Tap for more steps...

Step 3.15.7.11.5.1

To multiply absolute values, multiply the terms inside each absolute value.

$f'' (x) = \frac{\frac{2 (18 | (x^{2} - 6 x) (x^{2} - 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.5.2

Raise $x^{2} - 6 x$ to the power of $1$ .

Step 3.15.7.11.5.3

Raise $x^{2} - 6 x$ to the power of $1$ .

Step 3.15.7.11.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{2 (18 | {(x^{2} - 6 x)}^{1 + 1} | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.5.5

Add $1$ and $1$ .

$f'' (x) = \frac{\frac{2 (18 | {(x^{2} - 6 x)}^{2} | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.6

Rewrite ${(x^{2} - 6 x)}^{2}$ as $(x^{2} - 6 x) (x^{2} - 6 x)$ .

Step 3.15.7.11.7

Expand $(x^{2} - 6 x) (x^{2} - 6 x)$ using the FOIL Method.

Tap for more steps...

Step 3.15.7.11.7.1

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (18 | x^{2} (x^{2} - 6 x) - 6 x (x^{2} - 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.7.2

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (18 | x^{2} x^{2} + x^{2} (- 6 x) - 6 x (x^{2} - 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.7.3

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (18 | x^{2} x^{2} + x^{2} (- 6 x) - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8

Simplify and combine like terms.

Tap for more steps...

Step 3.15.7.11.8.1

Simplify each term.

Tap for more steps...

Step 3.15.7.11.8.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.8.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{2 (18 | x^{2 + 2} + x^{2} (- 6 x) - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} + x^{2} (- 6 x) - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{2} x - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.8.1.3.1

Move $x$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 (x \cdot x^{2}) - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.3.2

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 3.15.7.11.8.1.3.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.11.8.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{1 + 2} - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{3} - 6 x \cdot x^{2} - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.4

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.8.1.4.1

Move $x^{2}$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{3} - 6 (x^{2} x) - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.4.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 3.15.7.11.8.1.4.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.11.8.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{3} - 6 x^{2 + 1} - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.4.3

Add $2$ and $1$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{3} - 6 x^{3} - 6 x (- 6 x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{3} - 6 x^{3} - 6 \cdot (- 6 x \cdot x) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.6

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.8.1.6.1

Move $x$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{3} - 6 x^{3} - 6 \cdot (- 6 (x \cdot x)) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.6.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{3} - 6 x^{3} - 6 \cdot (- 6 x^{2}) | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.1.7

Multiply $- 6$ by $- 6$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 6 x^{3} - 6 x^{3} + 36 x^{2} | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.8.2

Subtract $6 x^{3}$ from $- 6 x^{3}$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | + x (- 2 x^{5} + 36 x^{4} - 234 x^{3} + 648 x^{2} + 3 x | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.9

Apply the distributive property.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | + x (- 2 x^{5}) + x (36 x^{4}) + x (- 234 x^{3}) + x (648 x^{2}) + x (3 x | x^{4} - 12 x^{3} + 36 x^{2} |) + x (- 648 x) + x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.10

Simplify.

Tap for more steps...

Step 3.15.7.11.10.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{5} + x (36 x^{4}) + x (- 234 x^{3}) + x (648 x^{2}) + x (3 x | x^{4} - 12 x^{3} + 36 x^{2} |) + x (- 648 x) + x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.10.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{5} + 36 x \cdot x^{4} + x (- 234 x^{3}) + x (648 x^{2}) + x (3 x | x^{4} - 12 x^{3} + 36 x^{2} |) + x (- 648 x) + x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.10.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{5} + 36 x \cdot x^{4} - 234 x \cdot x^{3} + x (648 x^{2}) + x (3 x | x^{4} - 12 x^{3} + 36 x^{2} |) + x (- 648 x) + x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.10.4

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{5} + 36 x \cdot x^{4} - 234 x \cdot x^{3} + 648 x \cdot x^{2} + x (3 x | x^{4} - 12 x^{3} + 36 x^{2} |) + x (- 648 x) + x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.10.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{5} + 36 x \cdot x^{4} - 234 x \cdot x^{3} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) + x (- 648 x) + x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.10.6

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{5} + 36 x \cdot x^{4} - 234 x \cdot x^{3} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x + x (- 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.10.7

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x \cdot x^{5} + 36 x \cdot x^{4} - 234 x \cdot x^{3} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11

Simplify each term.

Tap for more steps...

Step 3.15.7.11.11.1

Multiply $x$ by $x^{5}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.11.1.1

Move $x^{5}$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 (x^{5} x) + 36 x \cdot x^{4} - 234 x \cdot x^{3} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.1.2

Multiply $x^{5}$ by $x$ .

Tap for more steps...

Step 3.15.7.11.11.1.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.11.11.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{5 + 1} + 36 x \cdot x^{4} - 234 x \cdot x^{3} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.1.3

Add $5$ and $1$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x \cdot x^{4} - 234 x \cdot x^{3} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.2

Multiply $x$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.11.2.1

Move $x^{4}$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 (x^{4} x) - 234 x \cdot x^{3} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.2.2

Multiply $x^{4}$ by $x$ .

Tap for more steps...

Step 3.15.7.11.11.2.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.11.11.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{4 + 1} - 234 x \cdot x^{3} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.2.3

Add $4$ and $1$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x \cdot x^{3} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.3

Multiply $x$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.11.3.1

Move $x^{3}$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 (x^{3} x) + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.3.2

Multiply $x^{3}$ by $x$ .

Tap for more steps...

Step 3.15.7.11.11.3.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.11.11.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x^{3 + 1} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.3.3

Add $3$ and $1$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x^{4} + 648 x \cdot x^{2} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.4

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.11.4.1

Move $x^{2}$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x^{4} + 648 (x^{2} x) + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.4.2

Multiply $x^{2}$ by $x$ .

Tap for more steps...

Step 3.15.7.11.11.4.2.1

Raise $x$ to the power of $1$ .

Step 3.15.7.11.11.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x^{4} + 648 x^{2 + 1} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.4.3

Add $2$ and $1$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x^{4} + 648 x^{3} + 3 x (x | x^{4} - 12 x^{3} + 36 x^{2} |) - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.11.5.1

Move $x$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x^{4} + 648 x^{3} + 3 (x \cdot x) | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.5.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x^{4} + 648 x^{3} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x \cdot x - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.6

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 3.15.7.11.11.6.1

Move $x$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x^{4} + 648 x^{3} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 648 (x \cdot x) - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.11.6.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{\frac{2 (18 | x^{4} - 12 x^{3} + 36 x^{2} | - 2 x^{6} + 36 x^{5} - 234 x^{4} + 648 x^{3} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 648 x^{2} - 18 x | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.11.12

Reorder terms.

$f'' (x) = \frac{\frac{2 (- 2 x^{6} + 36 x^{5} - 234 x^{4} + 648 x^{3} - 648 x^{2} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.12

Factor $- 1$ out of $- 2 x^{6}$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6}) + 36 x^{5} - 234 x^{4} + 648 x^{3} - 648 x^{2} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.13

Factor $- 1$ out of $36 x^{5}$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6}) - (- 36 x^{5}) - 234 x^{4} + 648 x^{3} - 648 x^{2} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.14

Factor $- 1$ out of $- (2 x^{6}) - (- 36 x^{5})$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5}) - 234 x^{4} + 648 x^{3} - 648 x^{2} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.15

Factor $- 1$ out of $- 234 x^{4}$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5}) - (234 x^{4}) + 648 x^{3} - 648 x^{2} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.16

Factor $- 1$ out of $- (2 x^{6} - 36 x^{5}) - (234 x^{4})$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4}) + 648 x^{3} - 648 x^{2} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.17

Factor $- 1$ out of $648 x^{3}$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4}) - (- 648 x^{3}) - 648 x^{2} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.18

Factor $- 1$ out of $- (2 x^{6} - 36 x^{5} + 234 x^{4}) - (- 648 x^{3})$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3}) - 648 x^{2} + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.19

Factor $- 1$ out of $- 648 x^{2}$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3}) - (648 x^{2}) + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.20

Factor $- 1$ out of $- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3}) - (648 x^{2})$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2}) + 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.21

Factor $- 1$ out of $3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} |$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2}) - (- 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} |) - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.22

Factor $- 1$ out of $- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2}) - (- 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} |)$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} |) - 18 x | x^{4} - 12 x^{3} + 36 x^{2} | + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.23

Factor $- 1$ out of $- 18 x | x^{4} - 12 x^{3} + 36 x^{2} |$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} |) - (18 x | x^{4} - 12 x^{3} + 36 x^{2} |) + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.24

Factor $- 1$ out of $- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} |) - (18 x | x^{4} - 12 x^{3} + 36 x^{2} |)$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} |) + 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.25

Factor $- 1$ out of $18 | x^{4} - 12 x^{3} + 36 x^{2} |$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} |) - (- 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.26

Factor $- 1$ out of $- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} |) - (- 18 | x^{4} - 12 x^{3} + 36 x^{2} |)$ .

$f'' (x) = \frac{\frac{2 (- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} | - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.27

Rewrite $- (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} | - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)$ as $- 1 (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} | - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)$ .

$f'' (x) = \frac{\frac{2 (- 1 (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} | - 18 | x^{4} - 12 x^{3} + 36 x^{2} |))}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.7.28

Move the negative in front of the fraction.

$f'' (x) = \frac{- \frac{2 (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} | - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x \cdot x + x \cdot - 6 |}^{2}}$

Step 3.15.8

Combine terms.

Tap for more steps...

Step 3.15.8.1

Raise $x$ to the power of $1$ .

Step 3.15.8.2

Raise $x$ to the power of $1$ .

Step 3.15.8.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 3.15.8.4

Add $1$ and $1$ .

Step 3.15.8.5

Move $- 6$ to the left of $x$ .

Step 3.15.8.6

Rewrite $\frac{- \frac{2 (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} | - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}}{{| x^{2} - 6 x |}^{2}}$ as a product.

$f'' (x) = - \frac{2 (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} | - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |} \cdot \frac{1}{{| x^{2} - 6 x |}^{2}}$

Step 3.15.8.7

Multiply $\frac{1}{{| x^{2} - 6 x |}^{2}}$ by $\frac{2 (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} | - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{| x (x - 6) |}$ .

$f'' (x) = - \frac{2 (2 x^{6} - 36 x^{5} + 234 x^{4} - 648 x^{3} + 648 x^{2} - 3 x^{2} | x^{4} - 12 x^{3} + 36 x^{2} | + 18 x | x^{4} - 12 x^{3} + 36 x^{2} | - 18 | x^{4} - 12 x^{3} + 36 x^{2} |)}{{| x^{2} - 6 x |}^{2} | x (x - 6) |}$

Step 4

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |} = 0$

Step 5

Find the first derivative.

Tap for more steps...

Step 5.1

Find the first derivative.

Tap for more steps...

Step 5.1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = | x |$ and $g (x) = x^{2} - 6 x$ .

Tap for more steps...

Step 5.1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 6 x$ .

$f' (x) = \frac{d}{d u} (| u |) \frac{d}{d x} (x^{2} - 6 x)$

Step 5.1.1.2

The derivative of $| u |$ with respect to $u$ is $\frac{u}{| u |}$ .

$f' (x) = \frac{u}{| u |} \cdot \frac{d}{d x} (x^{2} - 6 x)$

Step 5.1.1.3

Replace all occurrences of $u$ with $x^{2} - 6 x$ .

$f' (x) = \frac{x^{2} - 6 x}{| x^{2} - 6 x |} \cdot \frac{d}{d x} (x^{2} - 6 x)$

Step 5.1.2

Differentiate.

Tap for more steps...

Step 5.1.2.1

By the Sum Rule, the derivative of $x^{2} - 6 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x]$ .

$f' (x) = \frac{x^{2} - 6 x}{| x^{2} - 6 x |} \cdot (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 6 x))$

Step 5.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = \frac{x^{2} - 6 x}{| x^{2} - 6 x |} \cdot (2 x + \frac{d}{d x} (- 6 x))$

Step 5.1.2.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$f' (x) = \frac{x^{2} - 6 x}{| x^{2} - 6 x |} \cdot (2 x - 6 \frac{d}{d x} x)$

Step 5.1.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = \frac{x^{2} - 6 x}{| x^{2} - 6 x |} \cdot (2 x - 6 \cdot 1)$

Step 5.1.2.5

Multiply $- 6$ by $1$ .

$f' (x) = \frac{x^{2} - 6 x}{| x^{2} - 6 x |} \cdot (2 x - 6)$

Step 5.1.3

Simplify.

Tap for more steps...

Step 5.1.3.1

Reorder the factors of $\frac{x^{2} - 6 x}{| x^{2} - 6 x |} (2 x - 6)$ .

$f' (x) = (2 x - 6) (\frac{x^{2} - 6 x}{| x^{2} - 6 x |})$

Step 5.1.3.2

Factor $x$ out of $x^{2} - 6 x$ .

Tap for more steps...

Step 5.1.3.2.1

Factor $x$ out of $x^{2}$ .

$f' (x) = (2 x - 6) (\frac{x \cdot x - 6 x}{| x^{2} - 6 x |})$

Step 5.1.3.2.2

Factor $x$ out of $- 6 x$ .

$f' (x) = (2 x - 6) (\frac{x \cdot x + x \cdot - 6}{| x^{2} - 6 x |})$

Step 5.1.3.2.3

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x^{2} - 6 x |})$

Step 5.1.3.3

Simplify the denominator.

Tap for more steps...

Step 5.1.3.3.1

Factor $x$ out of $x^{2} - 6 x$ .

Tap for more steps...

Step 5.1.3.3.1.1

Factor $x$ out of $x^{2}$ .

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x \cdot x - 6 x |})$

Step 5.1.3.3.1.2

Factor $x$ out of $- 6 x$ .

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x \cdot x + x \cdot - 6 |})$

Step 5.1.3.3.1.3

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x (x - 6) |})$

Step 5.1.3.3.2

Apply the distributive property.

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x \cdot x + x \cdot - 6 |})$

Step 5.1.3.3.3

Multiply $x$ by $x$ .

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x^{2} + x \cdot - 6 |})$

Step 5.1.3.3.4

Move $- 6$ to the left of $x$ .

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x^{2} - 6 \cdot x |})$

Step 5.1.3.3.5

Factor $x$ out of $x^{2} - 6 x$ .

Tap for more steps...

Step 5.1.3.3.5.1

Factor $x$ out of $x^{2}$ .

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x \cdot x - 6 x |})$

Step 5.1.3.3.5.2

Factor $x$ out of $- 6 x$ .

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x \cdot x + x \cdot - 6 |})$

Step 5.1.3.3.5.3

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$f' (x) = (2 x - 6) (\frac{x (x - 6)}{| x (x - 6) |})$

Step 5.1.3.4

Multiply $2 x - 6$ by $\frac{x (x - 6)}{| x (x - 6) |}$ .

$f' (x) = \frac{(2 x - 6) (x (x - 6))}{| x (x - 6) |}$

Step 5.1.3.5

Factor $2$ out of $2 x - 6$ .

Tap for more steps...

Step 5.1.3.5.1

Factor $2$ out of $2 x$ .

$f' (x) = \frac{(2 (x) - 6) x (x - 6)}{| x (x - 6) |}$

Step 5.1.3.5.2

Factor $2$ out of $- 6$ .

$f' (x) = \frac{(2 x + 2 \cdot - 3) x (x - 6)}{| x (x - 6) |}$

Step 5.1.3.5.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$f' (x) = \frac{2 (x - 3) x (x - 6)}{| x (x - 6) |}$

Step 5.1.3.6

Reorder factors in $\frac{2 (x - 3) x (x - 6)}{| x (x - 6) |}$ .

$f' (x) = \frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$ .

$\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |} = 0$ .

Tap for more steps...

Step 6.1

Set the first derivative equal to $0$ .

$\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |} = 0$

Step 6.2

Set the numerator equal to zero.

$2 x (x - 3) (x - 6) = 0$

Step 6.3

Solve the equation for $x$ .

Tap for more steps...

Step 6.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 3 = 0$

$x - 6 = 0$

Step 6.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 6.3.3

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.3.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 6.3.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6.3.4

Set $x - 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 6.3.4.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 6.3.4.2

Add $6$ to both sides of the equation.

$x = 6$

Step 6.3.5

The final solution is all the values that make $2 x (x - 3) (x - 6) = 0$ true.

$x = 0, 3, 6$

Step 6.4

Exclude the solutions that do not make $\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |} = 0$ true.

$x = 3$

Step 7

Find the values where the derivative is undefined.

Tap for more steps...

Step 7.1

Set the denominator in $\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$ equal to $0$ to find where the expression is undefined.

$| x (x - 6) | = 0$

Step 7.2

Solve for $x$ .

Tap for more steps...

Step 7.2.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x (x - 6) = \pm 0$

Step 7.2.2

Plus or minus $0$ is $0$ .

$x (x - 6) = 0$

Step 7.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 6 = 0$

Step 7.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 7.2.5

Set $x - 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 7.2.5.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 7.2.5.2

Add $6$ to both sides of the equation.

$x = 6$

Step 7.2.6

The final solution is all the values that make $x (x - 6) = 0$ true.

$x = 0, 6$

Step 7.3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 0, x = 6$

Step 8

Critical points to evaluate.

$x = 3, 0, 6$

Step 9

Evaluate the second derivative at $x = 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (2 {(3)}^{6} - 36 {(3)}^{5} + 234 {(3)}^{4} - 648 {(3)}^{3} + 648 {(3)}^{2} - 3 {(3)}^{2} | {(3)}^{4} - 12 {(3)}^{3} + 36 {(3)}^{2} | + 18 (3) | {(3)}^{4} - 12 {(3)}^{3} + 36 {(3)}^{2} | - 18 | {(3)}^{4} - 12 {(3)}^{3} + 36 {(3)}^{2} |)}{{| {(3)}^{2} - 6 \cdot 3 |}^{2} | (3) ((3) - 6) |}$

Step 10

Evaluate the second derivative.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Raise $3$ to the power of $6$ .

$- \frac{2 (2 \cdot 729 - 36 \cdot 3^{5} + 234 \cdot 3^{4} - 648 \cdot 3^{3} + 648 \cdot 3^{2} - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.2

Multiply $2$ by $729$ .

$- \frac{2 (1458 - 36 \cdot 3^{5} + 234 \cdot 3^{4} - 648 \cdot 3^{3} + 648 \cdot 3^{2} - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.3

Raise $3$ to the power of $5$ .

$- \frac{2 (1458 - 36 \cdot 243 + 234 \cdot 3^{4} - 648 \cdot 3^{3} + 648 \cdot 3^{2} - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.4

Multiply $- 36$ by $243$ .

$- \frac{2 (1458 - 8748 + 234 \cdot 3^{4} - 648 \cdot 3^{3} + 648 \cdot 3^{2} - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.5

Raise $3$ to the power of $4$ .

$- \frac{2 (1458 - 8748 + 234 \cdot 81 - 648 \cdot 3^{3} + 648 \cdot 3^{2} - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.6

Multiply $234$ by $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 648 \cdot 3^{3} + 648 \cdot 3^{2} - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.7

Raise $3$ to the power of $3$ .

$- \frac{2 (1458 - 8748 + 18954 - 648 \cdot 27 + 648 \cdot 3^{2} - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.8

Multiply $- 648$ by $27$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 648 \cdot 3^{2} - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.9

Raise $3$ to the power of $2$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 648 \cdot 9 - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.10

Multiply $648$ by $9$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 3 \cdot 3^{2} | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.11

Raise $3$ to the power of $2$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 3 \cdot 9 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.12

Multiply $- 3$ by $9$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 27 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.13

Simplify each term.

Tap for more steps...

Step 10.1.13.1

Raise $3$ to the power of $4$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 27 | 81 - 12 \cdot 3^{3} + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.13.2

Raise $3$ to the power of $3$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 27 | 81 - 12 \cdot 27 + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.13.3

Multiply $- 12$ by $27$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 27 | 81 - 324 + 36 \cdot 3^{2} | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.13.4

Raise $3$ to the power of $2$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 27 | 81 - 324 + 36 \cdot 9 | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.13.5

Multiply $36$ by $9$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 27 | 81 - 324 + 324 | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.14

Subtract $324$ from $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 27 | - 243 + 324 | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.15

Add $- 243$ and $324$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 27 | 81 | + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.16

The absolute value is the distance between a number and zero. The distance between $0$ and $81$ is $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 27 \cdot 81 + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.17

Multiply $- 27$ by $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 18 (3) | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.18

Multiply $18$ by $3$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 54 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.19

Simplify each term.

Tap for more steps...

Step 10.1.19.1

Raise $3$ to the power of $4$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 54 | 81 - 12 \cdot 3^{3} + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.19.2

Raise $3$ to the power of $3$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 54 | 81 - 12 \cdot 27 + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.19.3

Multiply $- 12$ by $27$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 54 | 81 - 324 + 36 \cdot 3^{2} | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.19.4

Raise $3$ to the power of $2$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 54 | 81 - 324 + 36 \cdot 9 | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.19.5

Multiply $36$ by $9$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 54 | 81 - 324 + 324 | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.20

Subtract $324$ from $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 54 | - 243 + 324 | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.21

Add $- 243$ and $324$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 54 | 81 | - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.22

The absolute value is the distance between a number and zero. The distance between $0$ and $81$ is $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 54 \cdot 81 - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.23

Multiply $54$ by $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 18 | 3^{4} - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.24

Simplify each term.

Tap for more steps...

Step 10.1.24.1

Raise $3$ to the power of $4$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 18 | 81 - 12 \cdot 3^{3} + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.24.2

Raise $3$ to the power of $3$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 18 | 81 - 12 \cdot 27 + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.24.3

Multiply $- 12$ by $27$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 18 | 81 - 324 + 36 \cdot 3^{2} |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.24.4

Raise $3$ to the power of $2$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 18 | 81 - 324 + 36 \cdot 9 |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.24.5

Multiply $36$ by $9$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 18 | 81 - 324 + 324 |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.25

Subtract $324$ from $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 18 | - 243 + 324 |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.26

Add $- 243$ and $324$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 18 | 81 |)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.27

The absolute value is the distance between a number and zero. The distance between $0$ and $81$ is $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 18 \cdot 81)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.28

Multiply $- 18$ by $81$ .

$- \frac{2 (1458 - 8748 + 18954 - 17496 + 5832 - 2187 + 4374 - 1458)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.29

Subtract $8748$ from $1458$ .

$- \frac{2 (- 7290 + 18954 - 17496 + 5832 - 2187 + 4374 - 1458)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.30

Add $- 7290$ and $18954$ .

$- \frac{2 (11664 - 17496 + 5832 - 2187 + 4374 - 1458)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.31

Subtract $17496$ from $11664$ .

$- \frac{2 (- 5832 + 5832 - 2187 + 4374 - 1458)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.32

Add $- 5832$ and $5832$ .

$- \frac{2 (0 - 2187 + 4374 - 1458)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.33

Subtract $2187$ from $0$ .

$- \frac{2 (- 2187 + 4374 - 1458)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.34

Add $- 2187$ and $4374$ .

$- \frac{2 (2187 - 1458)}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.1.35

Subtract $1458$ from $2187$ .

$- \frac{2 \cdot 729}{{| 3^{2} - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.2

Simplify the denominator.

Tap for more steps...

Step 10.2.1

Raise $3$ to the power of $2$ .

$- \frac{2 \cdot 729}{{| 9 - 6 \cdot 3 |}^{2} | 3 (3 - 6) |}$

Step 10.2.2

Multiply $- 6$ by $3$ .

$- \frac{2 \cdot 729}{{| 9 - 18 |}^{2} | 3 (3 - 6) |}$

Step 10.2.3

Subtract $18$ from $9$ .

$- \frac{2 \cdot 729}{{| - 9 |}^{2} | 3 (3 - 6) |}$

Step 10.2.4

Subtract $6$ from $3$ .

$- \frac{2 \cdot 729}{{| - 9 |}^{2} | 3 \cdot - 3 |}$

Step 10.2.5

The absolute value is the distance between a number and zero. The distance between $- 9$ and $0$ is $9$ .

$- \frac{2 \cdot 729}{9^{2} | 3 \cdot - 3 |}$

Step 10.2.6

Raise $9$ to the power of $2$ .

$- \frac{2 \cdot 729}{81 | 3 \cdot - 3 |}$

Step 10.2.7

Multiply $3$ by $- 3$ .

$- \frac{2 \cdot 729}{81 | - 9 |}$

Step 10.2.8

The absolute value is the distance between a number and zero. The distance between $- 9$ and $0$ is $9$ .

$- \frac{2 \cdot 729}{81 \cdot 9}$

Step 10.3

Simplify the expression.

Tap for more steps...

Step 10.3.1

Multiply $2$ by $729$ .

$- \frac{1458}{81 \cdot 9}$

Step 10.3.2

Multiply $81$ by $9$ .

$- \frac{1458}{729}$

Step 10.3.3

Divide $1458$ by $729$ .

$- 1 \cdot 2$

Step 10.3.4

Multiply $- 1$ by $2$ .

$- 2$

Step 11

$x = 3$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 3$ is a local maximum

Step 12

Find the y-value when $x = 3$ .

Tap for more steps...

Step 12.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = | {(3)}^{2} - 6 \cdot 3 |$

Step 12.2

Simplify the result.

Tap for more steps...

Step 12.2.1

Simplify each term.

Tap for more steps...

Step 12.2.1.1

Raise $3$ to the power of $2$ .

$f (3) = | 9 - 6 \cdot 3 |$

Step 12.2.1.2

Multiply $- 6$ by $3$ .

$f (3) = | 9 - 18 |$

Step 12.2.2

Subtract $18$ from $9$ .

$f (3) = | - 9 |$

Step 12.2.3

The absolute value is the distance between a number and zero. The distance between $- 9$ and $0$ is $9$ .

$f (3) = 9$

Step 12.2.4

The final answer is $9$ .

$y = 9$

Step 13

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{2 (2 {(0)}^{6} - 36 {(0)}^{5} + 234 {(0)}^{4} - 648 {(0)}^{3} + 648 {(0)}^{2} - 3 {(0)}^{2} | {(0)}^{4} - 12 {(0)}^{3} + 36 {(0)}^{2} | + 18 (0) | {(0)}^{4} - 12 {(0)}^{3} + 36 {(0)}^{2} | - 18 | {(0)}^{4} - 12 {(0)}^{3} + 36 {(0)}^{2} |)}{{| {(0)}^{2} - 6 \cdot 0 |}^{2} | (0) ((0) - 6) |}$

Step 14

Evaluate the second derivative.

Tap for more steps...

Step 14.1

Simplify each term.

Tap for more steps...

Step 14.1.1

Raising $0$ to any positive power yields $0$ .

Step 14.1.2

Multiply $- 6$ by $0$ .

Step 14.2

Add $0$ and $0$ .

Step 14.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

Step 14.4

Raising $0$ to any positive power yields $0$ .

Step 14.5

Subtract $6$ from $0$ .

Step 14.6

Multiply $0$ by $- 6$ .

Step 14.7

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

Step 14.8

Multiply $0$ by $0$ .

Step 14.9

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 15

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

Tap for more steps...

Step 15.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 3) \cup (3, 6) \cup (6, \infty)$

Step 15.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$ to check if the result is negative or positive.

Tap for more steps...

Step 15.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = \frac{2 (- 2) ((- 2) - 3) ((- 2) - 6)}{| (- 2) ((- 2) - 6) |}$

Step 15.2.2

Simplify the result.

Tap for more steps...

Step 15.2.2.1

Multiply $2$ by $- 2$ .

$f' (- 2) = \frac{- 4 (- 2 - 3) (- 2 - 6)}{| - 2 (- 2 - 6) |}$

Step 15.2.2.2

Simplify the denominator.

Tap for more steps...

Step 15.2.2.2.1

Subtract $6$ from $- 2$ .

$f' (- 2) = \frac{- 4 (- 2 - 3) (- 2 - 6)}{| - 2 \cdot - 8 |}$

Step 15.2.2.2.2

Multiply $- 2$ by $- 8$ .

$f' (- 2) = \frac{- 4 (- 2 - 3) (- 2 - 6)}{| 16 |}$

Step 15.2.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $16$ is $16$ .

$f' (- 2) = \frac{- 4 (- 2 - 3) (- 2 - 6)}{16}$

Step 15.2.2.3

Simplify the numerator.

Tap for more steps...

Step 15.2.2.3.1

Subtract $3$ from $- 2$ .

$f' (- 2) = \frac{- 4 \cdot (- 5 (- 2 - 6))}{16}$

Step 15.2.2.3.2

Multiply $- 4$ by $- 5$ .

$f' (- 2) = \frac{20 (- 2 - 6)}{16}$

Step 15.2.2.3.3

Subtract $6$ from $- 2$ .

$f' (- 2) = \frac{20 \cdot - 8}{16}$

Step 15.2.2.4

Simplify the expression.

Tap for more steps...

Step 15.2.2.4.1

Multiply $20$ by $- 8$ .

$f' (- 2) = \frac{- 160}{16}$

Step 15.2.2.4.2

Divide $- 160$ by $16$ .

$f' (- 2) = - 10$

Step 15.2.2.5

The final answer is $- 10$ .

$- 10$

Step 15.3

Substitute any number, such as $2$ , from the interval $(0, 3)$ in the first derivative $\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$ to check if the result is negative or positive.

Tap for more steps...

Step 15.3.1

Replace the variable $x$ with $2$ in the expression.

$f' (2) = \frac{2 (2) ((2) - 3) ((2) - 6)}{| (2) ((2) - 6) |}$

Step 15.3.2

Simplify the result.

Tap for more steps...

Step 15.3.2.1

Multiply $2$ by $2$ .

$f' (2) = \frac{4 (2 - 3) (2 - 6)}{| 2 (2 - 6) |}$

Step 15.3.2.2

Simplify the denominator.

Tap for more steps...

Step 15.3.2.2.1

Subtract $6$ from $2$ .

$f' (2) = \frac{4 (2 - 3) (2 - 6)}{| 2 \cdot - 4 |}$

Step 15.3.2.2.2

Multiply $2$ by $- 4$ .

$f' (2) = \frac{4 (2 - 3) (2 - 6)}{| - 8 |}$

Step 15.3.2.2.3

The absolute value is the distance between a number and zero. The distance between $- 8$ and $0$ is $8$ .

$f' (2) = \frac{4 (2 - 3) (2 - 6)}{8}$

Step 15.3.2.3

Simplify the numerator.

Tap for more steps...

Step 15.3.2.3.1

Subtract $3$ from $2$ .

$f' (2) = \frac{4 \cdot (- 1 (2 - 6))}{8}$

Step 15.3.2.3.2

Combine exponents.

Tap for more steps...

Step 15.3.2.3.2.1

Factor out negative.

$f' (2) = \frac{- (4 (2 - 6))}{8}$

Step 15.3.2.3.2.2

Multiply $4$ by $- 1$ .

$f' (2) = \frac{- 4 (2 - 6)}{8}$

Step 15.3.2.3.3

Subtract $6$ from $2$ .

$f' (2) = \frac{- 4 \cdot - 4}{8}$

Step 15.3.2.4

Simplify the expression.

Tap for more steps...

Step 15.3.2.4.1

Multiply $- 4$ by $- 4$ .

$f' (2) = \frac{16}{8}$

Step 15.3.2.4.2

Divide $16$ by $8$ .

$f' (2) = 2$

Step 15.3.2.5

The final answer is $2$ .

$2$

Step 15.4

Substitute any number, such as $4$ , from the interval $(3, 6)$ in the first derivative $\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$ to check if the result is negative or positive.

Tap for more steps...

Step 15.4.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = \frac{2 (4) ((4) - 3) ((4) - 6)}{| (4) ((4) - 6) |}$

Step 15.4.2

Simplify the result.

Tap for more steps...

Step 15.4.2.1

Multiply $2$ by $4$ .

$f' (4) = \frac{8 (4 - 3) (4 - 6)}{| 4 (4 - 6) |}$

Step 15.4.2.2

Simplify the denominator.

Tap for more steps...

Step 15.4.2.2.1

Subtract $6$ from $4$ .

$f' (4) = \frac{8 (4 - 3) (4 - 6)}{| 4 \cdot - 2 |}$

Step 15.4.2.2.2

Multiply $4$ by $- 2$ .

$f' (4) = \frac{8 (4 - 3) (4 - 6)}{| - 8 |}$

Step 15.4.2.2.3

The absolute value is the distance between a number and zero. The distance between $- 8$ and $0$ is $8$ .

$f' (4) = \frac{8 (4 - 3) (4 - 6)}{8}$

Step 15.4.2.3

Simplify the numerator.

Tap for more steps...

Step 15.4.2.3.1

Subtract $3$ from $4$ .

$f' (4) = \frac{8 \cdot (1 (4 - 6))}{8}$

Step 15.4.2.3.2

Multiply $8$ by $1$ .

$f' (4) = \frac{8 (4 - 6)}{8}$

Step 15.4.2.3.3

Subtract $6$ from $4$ .

$f' (4) = \frac{8 \cdot - 2}{8}$

Step 15.4.2.4

Simplify the expression.

Tap for more steps...

Step 15.4.2.4.1

Multiply $8$ by $- 2$ .

$f' (4) = \frac{- 16}{8}$

Step 15.4.2.4.2

Divide $- 16$ by $8$ .

$f' (4) = - 2$

Step 15.4.2.5

The final answer is $- 2$ .

$- 2$

Step 15.5

Substitute any number, such as $8$ , from the interval $(6, \infty)$ in the first derivative $\frac{2 x (x - 3) (x - 6)}{| x (x - 6) |}$ to check if the result is negative or positive.

Tap for more steps...

Step 15.5.1

Replace the variable $x$ with $8$ in the expression.

$f' (8) = \frac{2 (8) ((8) - 3) ((8) - 6)}{| (8) ((8) - 6) |}$

Step 15.5.2

Simplify the result.

Tap for more steps...

Step 15.5.2.1

Multiply $2$ by $8$ .

$f' (8) = \frac{16 (8 - 3) (8 - 6)}{| 8 (8 - 6) |}$

Step 15.5.2.2

Simplify the denominator.

Tap for more steps...

Step 15.5.2.2.1

Subtract $6$ from $8$ .

$f' (8) = \frac{16 (8 - 3) (8 - 6)}{| 8 \cdot 2 |}$

Step 15.5.2.2.2

Multiply $8$ by $2$ .

$f' (8) = \frac{16 (8 - 3) (8 - 6)}{| 16 |}$

Step 15.5.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $16$ is $16$ .

$f' (8) = \frac{16 (8 - 3) (8 - 6)}{16}$

Step 15.5.2.3

Simplify the numerator.

Tap for more steps...

Step 15.5.2.3.1

Subtract $3$ from $8$ .

$f' (8) = \frac{16 \cdot (5 (8 - 6))}{16}$

Step 15.5.2.3.2

Multiply $16$ by $5$ .

$f' (8) = \frac{80 (8 - 6)}{16}$

Step 15.5.2.3.3

Subtract $6$ from $8$ .

$f' (8) = \frac{80 \cdot 2}{16}$

Step 15.5.2.4

Simplify the expression.

Tap for more steps...

Step 15.5.2.4.1

Multiply $80$ by $2$ .

$f' (8) = \frac{160}{16}$

Step 15.5.2.4.2

Divide $160$ by $16$ .

$f' (8) = 10$

Step 15.5.2.5

The final answer is $10$ .

$10$

Step 15.6

Since the first derivative changed signs from negative to positive around $x = 0$ , then $x = 0$ is a local minimum.

$x = 0$ is a local minimum

Step 15.7

Since the first derivative changed signs from positive to negative around $x = 3$ , then $x = 3$ is a local maximum.

$x = 3$ is a local maximum

Step 15.8

Since the first derivative changed signs from negative to positive around $x = 6$ , then $x = 6$ is a local minimum.

$x = 6$ is a local minimum

Step 15.9

These are the local extrema for $f (x) = | x^{2} - 6 x |$ .

$x = 0$ is a local minimum

$x = 3$ is a local maximum

$x = 6$ is a local minimum

$x = 0$ is a local minimum

$x = 3$ is a local maximum

$x = 6$ is a local minimum

Step 16