Calculus Examples

$f (x) = x^{3} {(x - 2)}^{2} (x + 5)$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

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

$\frac{d}{d x} [x^{3} ((x - 2) (x - 2)) (x + 5)]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

Apply the distributive property.

$\frac{d}{d x} [x^{3} (x (x - 2) - 2 (x - 2)) (x + 5)]$

Step 1.2.2

Apply the distributive property.

$\frac{d}{d x} [x^{3} (x \cdot x + x \cdot - 2 - 2 (x - 2)) (x + 5)]$

Step 1.2.3

Apply the distributive property.

$\frac{d}{d x} [x^{3} (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) (x + 5)]$

Step 1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.3.1

Simplify each term.

Tap for more steps...

Step 1.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{3} (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) (x + 5)]$

Step 1.3.1.2

Move $- 2$ to the left of $x$ .

$\frac{d}{d x} [x^{3} (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) (x + 5)]$

Step 1.3.1.3

Multiply $- 2$ by $- 2$ .

$\frac{d}{d x} [x^{3} (x^{2} - 2 x - 2 x + 4) (x + 5)]$

Step 1.3.2

Subtract $2 x$ from $- 2 x$ .

$\frac{d}{d x} [x^{3} (x^{2} - 4 x + 4) (x + 5)]$

Step 1.4

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 $f (x) = x^{3} (x^{2} - 4 x + 4)$ and $g (x) = x + 5$ .

$x^{3} (x^{2} - 4 x + 4) \frac{d}{d x} [x + 5] + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 1.5

Differentiate.

Tap for more steps...

Step 1.5.1

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$x^{3} (x^{2} - 4 x + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [5]) + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 1.5.2

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

$x^{3} (x^{2} - 4 x + 4) (1 + \frac{d}{d x} [5]) + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 1.5.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$x^{3} (x^{2} - 4 x + 4) (1 + 0) + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 1.5.4

Simplify the expression.

Tap for more steps...

Step 1.5.4.1

Add $1$ and $0$ .

$x^{3} (x^{2} - 4 x + 4) \cdot 1 + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 1.5.4.2

Multiply $x^{3}$ by $1$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 1.6

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 $f (x) = x^{3}$ and $g (x) = x^{2} - 4 x + 4$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} \frac{d}{d x} [x^{2} - 4 x + 4] + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.7

Differentiate.

Tap for more steps...

Step 1.7.1

By the Sum Rule, the derivative of $x^{2} - 4 x + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4]$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.7.2

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

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.7.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.7.4

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

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4 \cdot 1 + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.7.5

Multiply $- 4$ by $1$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4 + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.7.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4 + 0) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.7.7

Add $2 x - 4$ and $0$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.7.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4) + (x^{2} - 4 x + 4) (3 x^{2}))$

Step 1.7.9

Move $3$ to the left of $x^{2} - 4 x + 4$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4) + 3 \cdot (x^{2} - 4 x + 4) x^{2})$

Step 1.8

Simplify.

Tap for more steps...

Step 1.8.1

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x - 4) + 3 (x^{2} - 4 x + 4) x^{2})$

Step 1.8.2

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x) + x^{3} \cdot - 4 + 3 (x^{2} - 4 x + 4) x^{2})$

Step 1.8.3

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x) + x^{3} \cdot - 4 + (3 x^{2} + 3 (- 4 x) + 3 \cdot 4) x^{2})$

Step 1.8.4

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x) + x^{3} \cdot - 4 + 3 x^{2} x^{2} + 3 (- 4 x) x^{2} + 3 \cdot 4 x^{2})$

Step 1.8.5

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x)) + (x + 5) (x^{3} \cdot - 4) + (x + 5) (3 x^{2} x^{2}) + (x + 5) (3 (- 4 x) x^{2}) + (x + 5) (3 \cdot 4 x^{2})$

Step 1.8.6

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + (x + 5) (x^{3} \cdot - 4) + (x + 5) (3 x^{2} x^{2}) + (x + 5) (3 (- 4 x) x^{2}) + (x + 5) (3 \cdot 4 x^{2})$

Step 1.8.7

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + (x + 5) (3 x^{2} x^{2}) + (x + 5) (3 (- 4 x) x^{2}) + (x + 5) (3 \cdot 4 x^{2})$

Step 1.8.8

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + (x + 5) (3 (- 4 x) x^{2}) + (x + 5) (3 \cdot 4 x^{2})$

Step 1.8.9

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + (x + 5) (3 \cdot 4 x^{2})$

Step 1.8.10

Apply the distributive property.

Step 1.8.11

Combine terms.

Tap for more steps...

Step 1.8.11.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.8.11.1.1

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

$x^{3 + 2} + x^{3} (- 4 x) + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.1.2

Add $3$ and $2$ .

$x^{5} + x^{3} (- 4 x) + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.2

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

Tap for more steps...

Step 1.8.11.2.1

Move $x$ .

$x^{5} + x \cdot x^{3} \cdot - 4 + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.2.2

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

Tap for more steps...

Step 1.8.11.2.2.1

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

$x^{5} + x^{1} x^{3} \cdot - 4 + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.2.2.2

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

$x^{5} + x^{1 + 3} \cdot - 4 + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.2.3

Add $1$ and $3$ .

$x^{5} + x^{4} \cdot - 4 + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.3

Move $- 4$ to the left of $x^{4}$ .

$x^{5} - 4 \cdot x^{4} + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.4

Move $4$ to the left of $x^{3}$ .

$x^{5} - 4 x^{4} + 4 \cdot x^{3} + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.5

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

Tap for more steps...

Step 1.8.11.5.1

Move $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + x (x \cdot x^{3} \cdot 2) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.5.2

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

Tap for more steps...

Step 1.8.11.5.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + x (x^{1} x^{3} \cdot 2) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.5.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + x (x^{1 + 3} \cdot 2) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.5.3

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + x (x^{4} \cdot 2) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.6

Move $2$ to the left of $x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + x (2 \cdot x^{4}) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.7

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 (x^{1} x^{4}) + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.8

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{1 + 4} + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.9

Add $1$ and $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 5 (x^{3} (2 x)) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.10

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

Tap for more steps...

Step 1.8.11.10.1

Move $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 5 (x \cdot x^{3} \cdot 2) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.10.2

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

Tap for more steps...

Step 1.8.11.10.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 5 (x^{1} x^{3} \cdot 2) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.10.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 5 (x^{1 + 3} \cdot 2) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.10.3

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 5 (x^{4} \cdot 2) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.11

Move $2$ to the left of $x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 5 (2 \cdot x^{4}) + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.12

Multiply $2$ by $5$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 10 x^{4} + x (x^{3} \cdot - 4) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.13

Move $- 4$ to the left of $x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 10 x^{4} + x (- 4 \cdot x^{3}) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.14

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 10 x^{4} - 4 (x^{1} x^{3}) + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.15

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 10 x^{4} - 4 x^{1 + 3} + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.16

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 10 x^{4} - 4 x^{4} + 5 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.17

Move $- 4$ to the left of $x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 10 x^{4} - 4 x^{4} + 5 (- 4 \cdot x^{3}) + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.18

Multiply $- 4$ by $5$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 10 x^{4} - 4 x^{4} - 20 x^{3} + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.19

Subtract $4 x^{4}$ from $10 x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.20

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

Tap for more steps...

Step 1.8.11.20.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + x (3 (x^{2} x^{2})) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.20.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + x (3 x^{2 + 2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.20.3

Add $2$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + x (3 x^{4}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.21

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 (x^{1} x^{4}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.22

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{1 + 4} + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.23

Add $1$ and $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.24

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

Tap for more steps...

Step 1.8.11.24.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 5 (3 (x^{2} x^{2})) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.24.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 5 (3 x^{2 + 2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.24.3

Add $2$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 5 (3 x^{4}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.25

Multiply $3$ by $5$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.26

Multiply $- 4$ by $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 x \cdot x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.27

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

Tap for more steps...

Step 1.8.11.27.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 (x^{2} x)) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.27.2

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

Tap for more steps...

Step 1.8.11.27.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 (x^{2} x^{1})) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.27.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 x^{2 + 1}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.27.3

Add $2$ and $1$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 x^{3}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.28

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 (x^{1} x^{3}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.29

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{1 + 3} + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.30

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.31

Multiply $- 4$ by $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 x \cdot x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.32

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

Tap for more steps...

Step 1.8.11.32.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 (x^{2} x)) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.32.2

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

Tap for more steps...

Step 1.8.11.32.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 (x^{2} x^{1})) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.32.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 x^{2 + 1}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.32.3

Add $2$ and $1$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 x^{3}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.33

Multiply $- 12$ by $5$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} - 60 x^{3} + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.34

Subtract $12 x^{4}$ from $15 x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.35

Multiply $3$ by $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + x (12 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.36

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 (x^{1} x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.37

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 x^{1 + 2} + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.38

Add $1$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 x^{3} + 5 (3 \cdot 4 x^{2})$

Step 1.8.11.39

Multiply $3$ by $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 x^{3} + 5 (12 x^{2})$

Step 1.8.11.40

Multiply $12$ by $5$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 x^{3} + 60 x^{2}$

Step 1.8.11.41

Add $- 60 x^{3}$ and $12 x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 48 x^{3} + 60 x^{2}$

Step 1.8.11.42

Add $2 x^{5}$ and $3 x^{5}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 5 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{4} - 48 x^{3} + 60 x^{2}$

Step 1.8.11.43

Add $6 x^{4}$ and $3 x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 5 x^{5} + 9 x^{4} - 20 x^{3} - 48 x^{3} + 60 x^{2}$

Step 1.8.11.44

Subtract $48 x^{3}$ from $- 20 x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 5 x^{5} + 9 x^{4} - 68 x^{3} + 60 x^{2}$

Step 1.8.11.45

Add $x^{5}$ and $5 x^{5}$ .

$6 x^{5} - 4 x^{4} + 4 x^{3} + 9 x^{4} - 68 x^{3} + 60 x^{2}$

Step 1.8.11.46

Add $- 4 x^{4}$ and $9 x^{4}$ .

$6 x^{5} + 5 x^{4} + 4 x^{3} - 68 x^{3} + 60 x^{2}$

Step 1.8.11.47

Subtract $68 x^{3}$ from $4 x^{3}$ .

$6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$ with respect to $x$ is $\frac{d}{d x} [6 x^{5}] + \frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- 64 x^{3}] + \frac{d}{d x} [60 x^{2}]$ .

$f'' (x) = \frac{d}{d x} (6 x^{5}) + \frac{d}{d x} (5 x^{4}) + \frac{d}{d x} (- 64 x^{3}) + \frac{d}{d x} (60 x^{2})$

Step 2.2

Evaluate $\frac{d}{d x} [6 x^{5}]$ .

Tap for more steps...

Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{5}$ with respect to $x$ is $6 \frac{d}{d x} [x^{5}]$ .

$f'' (x) = 6 \frac{d}{d x} (x^{5}) + \frac{d}{d x} (5 x^{4}) + \frac{d}{d x} (- 64 x^{3}) + \frac{d}{d x} (60 x^{2})$

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$f'' (x) = 6 (5 x^{4}) + \frac{d}{d x} (5 x^{4}) + \frac{d}{d x} (- 64 x^{3}) + \frac{d}{d x} (60 x^{2})$

Step 2.2.3

Multiply $5$ by $6$ .

$f'' (x) = 30 x^{4} + \frac{d}{d x} (5 x^{4}) + \frac{d}{d x} (- 64 x^{3}) + \frac{d}{d x} (60 x^{2})$

Step 2.3

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

Tap for more steps...

Step 2.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$f'' (x) = 30 x^{4} + 5 \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 64 x^{3}) + \frac{d}{d x} (60 x^{2})$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f'' (x) = 30 x^{4} + 5 (4 x^{3}) + \frac{d}{d x} (- 64 x^{3}) + \frac{d}{d x} (60 x^{2})$

Step 2.3.3

Multiply $4$ by $5$ .

$f'' (x) = 30 x^{4} + 20 x^{3} + \frac{d}{d x} (- 64 x^{3}) + \frac{d}{d x} (60 x^{2})$

Step 2.4

Evaluate $\frac{d}{d x} [- 64 x^{3}]$ .

Tap for more steps...

Step 2.4.1

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64 x^{3}$ with respect to $x$ is $- 64 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 30 x^{4} + 20 x^{3} - 64 \frac{d}{d x} x^{3} + \frac{d}{d x} (60 x^{2})$

Step 2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 30 x^{4} + 20 x^{3} - 64 (3 x^{2}) + \frac{d}{d x} (60 x^{2})$

Step 2.4.3

Multiply $3$ by $- 64$ .

$f'' (x) = 30 x^{4} + 20 x^{3} - 192 x^{2} + \frac{d}{d x} (60 x^{2})$

Step 2.5

Evaluate $\frac{d}{d x} [60 x^{2}]$ .

Tap for more steps...

Step 2.5.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x^{2}$ with respect to $x$ is $60 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 30 x^{4} + 20 x^{3} - 192 x^{2} + 60 \frac{d}{d x} (x^{2})$

Step 2.5.2

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

$f'' (x) = 30 x^{4} + 20 x^{3} - 192 x^{2} + 60 (2 x)$

Step 2.5.3

Multiply $2$ by $60$ .

$f'' (x) = 30 x^{4} + 20 x^{3} - 192 x^{2} + 120 x$

Step 3

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

$6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2} = 0$

Step 4

Find the first derivative.

Tap for more steps...

Step 4.1

Find the first derivative.

Tap for more steps...

Step 4.1.1

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

$\frac{d}{d x} [x^{3} ((x - 2) (x - 2)) (x + 5)]$

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

Apply the distributive property.

$\frac{d}{d x} [x^{3} (x (x - 2) - 2 (x - 2)) (x + 5)]$

Step 4.1.2.2

Apply the distributive property.

$\frac{d}{d x} [x^{3} (x \cdot x + x \cdot - 2 - 2 (x - 2)) (x + 5)]$

Step 4.1.2.3

Apply the distributive property.

$\frac{d}{d x} [x^{3} (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) (x + 5)]$

Step 4.1.3

Simplify and combine like terms.

Tap for more steps...

Step 4.1.3.1

Simplify each term.

Tap for more steps...

Step 4.1.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [x^{3} (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) (x + 5)]$

Step 4.1.3.1.2

Move $- 2$ to the left of $x$ .

$\frac{d}{d x} [x^{3} (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) (x + 5)]$

Step 4.1.3.1.3

Multiply $- 2$ by $- 2$ .

$\frac{d}{d x} [x^{3} (x^{2} - 2 x - 2 x + 4) (x + 5)]$

Step 4.1.3.2

Subtract $2 x$ from $- 2 x$ .

$\frac{d}{d x} [x^{3} (x^{2} - 4 x + 4) (x + 5)]$

Step 4.1.4

$x^{3} (x^{2} - 4 x + 4) \frac{d}{d x} [x + 5] + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 4.1.5

Differentiate.

Tap for more steps...

Step 4.1.5.1

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$x^{3} (x^{2} - 4 x + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [5]) + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 4.1.5.2

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

$x^{3} (x^{2} - 4 x + 4) (1 + \frac{d}{d x} [5]) + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 4.1.5.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$x^{3} (x^{2} - 4 x + 4) (1 + 0) + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 4.1.5.4

Simplify the expression.

Tap for more steps...

Step 4.1.5.4.1

Add $1$ and $0$ .

$x^{3} (x^{2} - 4 x + 4) \cdot 1 + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 4.1.5.4.2

Multiply $x^{3}$ by $1$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 4.1.6

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} \frac{d}{d x} [x^{2} - 4 x + 4] + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 4.1.7

Differentiate.

Tap for more steps...

Step 4.1.7.1

By the Sum Rule, the derivative of $x^{2} - 4 x + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4]$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 4.1.7.2

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

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 4.1.7.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 4.1.7.4

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

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4 \cdot 1 + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 4.1.7.5

Multiply $- 4$ by $1$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4 + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 4.1.7.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4 + 0) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 4.1.7.7

Add $2 x - 4$ and $0$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 4.1.7.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4) + (x^{2} - 4 x + 4) (3 x^{2}))$

Step 4.1.7.9

Move $3$ to the left of $x^{2} - 4 x + 4$ .

$x^{3} (x^{2} - 4 x + 4) + (x + 5) (x^{3} (2 x - 4) + 3 \cdot (x^{2} - 4 x + 4) x^{2})$

Step 4.1.8

Simplify.

Tap for more steps...

Step 4.1.8.1

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x - 4) + 3 (x^{2} - 4 x + 4) x^{2})$

Step 4.1.8.2

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x) + x^{3} \cdot - 4 + 3 (x^{2} - 4 x + 4) x^{2})$

Step 4.1.8.3

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x) + x^{3} \cdot - 4 + (3 x^{2} + 3 (- 4 x) + 3 \cdot 4) x^{2})$

Step 4.1.8.4

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x) + x^{3} \cdot - 4 + 3 x^{2} x^{2} + 3 (- 4 x) x^{2} + 3 \cdot 4 x^{2})$

Step 4.1.8.5

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + (x + 5) (x^{3} (2 x)) + (x + 5) (x^{3} \cdot - 4) + (x + 5) (3 x^{2} x^{2}) + (x + 5) (3 (- 4 x) x^{2}) + (x + 5) (3 \cdot 4 x^{2})$

Step 4.1.8.6

Apply the distributive property.

$x^{3} x^{2} + x^{3} (- 4 x) + x^{3} \cdot 4 + x (x^{3} (2 x)) + 5 (x^{3} (2 x)) + (x + 5) (x^{3} \cdot - 4) + (x + 5) (3 x^{2} x^{2}) + (x + 5) (3 (- 4 x) x^{2}) + (x + 5) (3 \cdot 4 x^{2})$

Step 4.1.8.7

Apply the distributive property.

Step 4.1.8.8

Apply the distributive property.

Step 4.1.8.9

Apply the distributive property.

Step 4.1.8.10

Apply the distributive property.

Step 4.1.8.11

Combine terms.

Tap for more steps...

Step 4.1.8.11.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 4.1.8.11.1.1

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

Step 4.1.8.11.1.2

Add $3$ and $2$ .

Step 4.1.8.11.2

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

Tap for more steps...

Step 4.1.8.11.2.1

Move $x$ .

Step 4.1.8.11.2.2

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

Tap for more steps...

Step 4.1.8.11.2.2.1

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

Step 4.1.8.11.2.2.2

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

Step 4.1.8.11.2.3

Add $1$ and $3$ .

Step 4.1.8.11.3

Move $- 4$ to the left of $x^{4}$ .

Step 4.1.8.11.4

Move $4$ to the left of $x^{3}$ .

Step 4.1.8.11.5

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

Tap for more steps...

Step 4.1.8.11.5.1

Move $x$ .

Step 4.1.8.11.5.2

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

Tap for more steps...

Step 4.1.8.11.5.2.1

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

Step 4.1.8.11.5.2.2

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

Step 4.1.8.11.5.3

Add $1$ and $3$ .

Step 4.1.8.11.6

Move $2$ to the left of $x^{4}$ .

Step 4.1.8.11.7

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

Step 4.1.8.11.8

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

Step 4.1.8.11.9

Add $1$ and $4$ .

Step 4.1.8.11.10

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

Tap for more steps...

Step 4.1.8.11.10.1

Move $x$ .

Step 4.1.8.11.10.2

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

Tap for more steps...

Step 4.1.8.11.10.2.1

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

Step 4.1.8.11.10.2.2

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

Step 4.1.8.11.10.3

Add $1$ and $3$ .

Step 4.1.8.11.11

Move $2$ to the left of $x^{4}$ .

Step 4.1.8.11.12

Multiply $2$ by $5$ .

Step 4.1.8.11.13

Move $- 4$ to the left of $x^{3}$ .

Step 4.1.8.11.14

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

Step 4.1.8.11.15

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

Step 4.1.8.11.16

Add $1$ and $3$ .

Step 4.1.8.11.17

Move $- 4$ to the left of $x^{3}$ .

Step 4.1.8.11.18

Multiply $- 4$ by $5$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 10 x^{4} - 4 x^{4} - 20 x^{3} + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.19

Subtract $4 x^{4}$ from $10 x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + x (3 x^{2} x^{2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.20

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

Tap for more steps...

Step 4.1.8.11.20.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + x (3 (x^{2} x^{2})) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.20.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + x (3 x^{2 + 2}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.20.3

Add $2$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + x (3 x^{4}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.21

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 (x^{1} x^{4}) + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.22

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{1 + 4} + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.23

Add $1$ and $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 5 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.24

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

Tap for more steps...

Step 4.1.8.11.24.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 5 (3 (x^{2} x^{2})) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.24.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 5 (3 x^{2 + 2}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.24.3

Add $2$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 5 (3 x^{4}) + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.25

Multiply $3$ by $5$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (3 (- 4 x) x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.26

Multiply $- 4$ by $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 x \cdot x^{2}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.27

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

Tap for more steps...

Step 4.1.8.11.27.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 (x^{2} x)) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.27.2

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

Tap for more steps...

Step 4.1.8.11.27.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 (x^{2} x^{1})) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.27.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 x^{2 + 1}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.27.3

Add $2$ and $1$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} + x (- 12 x^{3}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.28

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 (x^{1} x^{3}) + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.29

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{1 + 3} + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.30

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.31

Multiply $- 4$ by $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 x \cdot x^{2}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.32

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

Tap for more steps...

Step 4.1.8.11.32.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 (x^{2} x)) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.32.2

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

Tap for more steps...

Step 4.1.8.11.32.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 (x^{2} x^{1})) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.32.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 x^{2 + 1}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.32.3

Add $2$ and $1$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} + 5 (- 12 x^{3}) + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.33

Multiply $- 12$ by $5$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 15 x^{4} - 12 x^{4} - 60 x^{3} + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.34

Subtract $12 x^{4}$ from $15 x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + x (3 \cdot 4 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.35

Multiply $3$ by $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + x (12 x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.36

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 (x^{1} x^{2}) + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.37

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 x^{1 + 2} + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.38

Add $1$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 x^{3} + 5 (3 \cdot 4 x^{2})$

Step 4.1.8.11.39

Multiply $3$ by $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 x^{3} + 5 (12 x^{2})$

Step 4.1.8.11.40

Multiply $12$ by $5$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 60 x^{3} + 12 x^{3} + 60 x^{2}$

Step 4.1.8.11.41

Add $- 60 x^{3}$ and $12 x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{5} + 3 x^{4} - 48 x^{3} + 60 x^{2}$

Step 4.1.8.11.42

Add $2 x^{5}$ and $3 x^{5}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 5 x^{5} + 6 x^{4} - 20 x^{3} + 3 x^{4} - 48 x^{3} + 60 x^{2}$

Step 4.1.8.11.43

Add $6 x^{4}$ and $3 x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 5 x^{5} + 9 x^{4} - 20 x^{3} - 48 x^{3} + 60 x^{2}$

Step 4.1.8.11.44

Subtract $48 x^{3}$ from $- 20 x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 5 x^{5} + 9 x^{4} - 68 x^{3} + 60 x^{2}$

Step 4.1.8.11.45

Add $x^{5}$ and $5 x^{5}$ .

$6 x^{5} - 4 x^{4} + 4 x^{3} + 9 x^{4} - 68 x^{3} + 60 x^{2}$

Step 4.1.8.11.46

Add $- 4 x^{4}$ and $9 x^{4}$ .

$6 x^{5} + 5 x^{4} + 4 x^{3} - 68 x^{3} + 60 x^{2}$

Step 4.1.8.11.47

Subtract $68 x^{3}$ from $4 x^{3}$ .

$f' (x) = 6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$ .

$6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$

Step 5

Set the first derivative equal to $0$ then solve the equation $6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2} = 0$ .

Tap for more steps...

Step 5.1

Set the first derivative equal to $0$ .

$6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2} = 0$

Step 5.2

Factor the left side of the equation.

Tap for more steps...

Step 5.2.1

Factor $x^{2}$ out of $6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$ .

Tap for more steps...

Step 5.2.1.1

Factor $x^{2}$ out of $6 x^{5}$ .

$x^{2} (6 x^{3}) + 5 x^{4} - 64 x^{3} + 60 x^{2} = 0$

Step 5.2.1.2

Factor $x^{2}$ out of $5 x^{4}$ .

$x^{2} (6 x^{3}) + x^{2} (5 x^{2}) - 64 x^{3} + 60 x^{2} = 0$

Step 5.2.1.3

Factor $x^{2}$ out of $- 64 x^{3}$ .

$x^{2} (6 x^{3}) + x^{2} (5 x^{2}) + x^{2} (- 64 x) + 60 x^{2} = 0$

Step 5.2.1.4

Factor $x^{2}$ out of $60 x^{2}$ .

$x^{2} (6 x^{3}) + x^{2} (5 x^{2}) + x^{2} (- 64 x) + x^{2} \cdot 60 = 0$

Step 5.2.1.5

Factor $x^{2}$ out of $x^{2} (6 x^{3}) + x^{2} (5 x^{2})$ .

$x^{2} (6 x^{3} + 5 x^{2}) + x^{2} (- 64 x) + x^{2} \cdot 60 = 0$

Step 5.2.1.6

Factor $x^{2}$ out of $x^{2} (6 x^{3} + 5 x^{2}) + x^{2} (- 64 x)$ .

$x^{2} (6 x^{3} + 5 x^{2} - 64 x) + x^{2} \cdot 60 = 0$

Step 5.2.1.7

Factor $x^{2}$ out of $x^{2} (6 x^{3} + 5 x^{2} - 64 x) + x^{2} \cdot 60$ .

$x^{2} (6 x^{3} + 5 x^{2} - 64 x + 60) = 0$

Step 5.2.2

Factor.

Tap for more steps...

Step 5.2.2.1

Factor $6 x^{3} + 5 x^{2} - 64 x + 60$ using the rational roots test.

Tap for more steps...

Step 5.2.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 60, \pm 2, \pm 30, \pm 3, \pm 20, \pm 4, \pm 15, \pm 5, \pm 12, \pm 6, \pm 10$

$q = \pm 1, \pm 6, \pm 2, \pm 3$

Step 5.2.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.1\overline{6}, \pm 0.5, \pm 0.\overline{3}, \pm 60, \pm 10, \pm 30, \pm 20, \pm 2, \pm 0.\overline{6}, \pm 5, \pm 15, \pm 3, \pm 1.5, \pm 3.\overline{3}, \pm 6.\overline{6}, \pm 4, \pm 1.\overline{3}, \pm 2.5, \pm 7.5, \pm 0.8\overline{3}, \pm 1.\overline{6}, \pm 12, \pm 6$

Step 5.2.2.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Tap for more steps...

Step 5.2.2.1.3.1

Substitute $2$ into the polynomial.

$6 \cdot 2^{3} + 5 \cdot 2^{2} - 64 \cdot 2 + 60$

Step 5.2.2.1.3.2

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

$6 \cdot 8 + 5 \cdot 2^{2} - 64 \cdot 2 + 60$

Step 5.2.2.1.3.3

Multiply $6$ by $8$ .

$48 + 5 \cdot 2^{2} - 64 \cdot 2 + 60$

Step 5.2.2.1.3.4

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

$48 + 5 \cdot 4 - 64 \cdot 2 + 60$

Step 5.2.2.1.3.5

Multiply $5$ by $4$ .

$48 + 20 - 64 \cdot 2 + 60$

Step 5.2.2.1.3.6

Add $48$ and $20$ .

$68 - 64 \cdot 2 + 60$

Step 5.2.2.1.3.7

Multiply $- 64$ by $2$ .

$68 - 128 + 60$

Step 5.2.2.1.3.8

Subtract $128$ from $68$ .

$- 60 + 60$

Step 5.2.2.1.3.9

Add $- 60$ and $60$ .

$0$

Step 5.2.2.1.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{6 x^{3} + 5 x^{2} - 64 x + 60}{x - 2}$

Step 5.2.2.1.5

Divide $6 x^{3} + 5 x^{2} - 64 x + 60$ by $x - 2$ .

Tap for more steps...

Step 5.2.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$6 x^{3}$

$5 x^{2}$

$64 x$

$60$

Step 5.2.2.1.5.2

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $x$ .

					$6 x^{2}$
	$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$

Step 5.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$6 x^{2}$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			+	$6 x^{3}$	-	$12 x^{2}$

Step 5.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} - 12 x^{2}$

				$6 x^{2}$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$

Step 5.2.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$6 x^{2}$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$

Step 5.2.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$6 x^{2}$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$

Step 5.2.2.1.5.7

Divide the highest order term in the dividend $17 x^{2}$ by the highest order term in divisor $x$ .

				$6 x^{2}$	+	$17 x$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$

Step 5.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$6 x^{2}$	+	$17 x$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$
					+	$17 x^{2}$	-	$34 x$

Step 5.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $17 x^{2} - 34 x$

				$6 x^{2}$	+	$17 x$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$
					-	$17 x^{2}$	+	$34 x$

Step 5.2.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$6 x^{2}$	+	$17 x$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$
					-	$17 x^{2}$	+	$34 x$
							-	$30 x$

Step 5.2.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$6 x^{2}$	+	$17 x$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$
					-	$17 x^{2}$	+	$34 x$
							-	$30 x$	+	$60$

Step 5.2.2.1.5.12

Divide the highest order term in the dividend $- 30 x$ by the highest order term in divisor $x$ .

				$6 x^{2}$	+	$17 x$	-	$30$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$
					-	$17 x^{2}$	+	$34 x$
							-	$30 x$	+	$60$

Step 5.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$6 x^{2}$	+	$17 x$	-	$30$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$
					-	$17 x^{2}$	+	$34 x$
							-	$30 x$	+	$60$
							-	$30 x$	+	$60$

Step 5.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 30 x + 60$

				$6 x^{2}$	+	$17 x$	-	$30$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$
					-	$17 x^{2}$	+	$34 x$
							-	$30 x$	+	$60$
							+	$30 x$	-	$60$

Step 5.2.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$6 x^{2}$	+	$17 x$	-	$30$
$x$	-	$2$		$6 x^{3}$	+	$5 x^{2}$	-	$64 x$	+	$60$
			-	$6 x^{3}$	+	$12 x^{2}$
					+	$17 x^{2}$	-	$64 x$
					-	$17 x^{2}$	+	$34 x$
							-	$30 x$	+	$60$
							+	$30 x$	-	$60$
										$0$

Step 5.2.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$6 x^{2} + 17 x - 30$

Step 5.2.2.1.6

Write $6 x^{3} + 5 x^{2} - 64 x + 60$ as a set of factors.

$x^{2} ((x - 2) (6 x^{2} + 17 x - 30)) = 0$

Step 5.2.2.2

Remove unnecessary parentheses.

$x^{2} (x - 2) (6 x^{2} + 17 x - 30) = 0$

Step 5.3

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

$x^{2} = 0$

$x - 2 = 0$

$6 x^{2} + 17 x - 30 = 0$

Step 5.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 5.4.2

Solve $x^{2} = 0$ for $x$ .

Tap for more steps...

Step 5.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 5.4.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 5.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 5.4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 5.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.5

Set $x - 2$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 5.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5.6

Set $6 x^{2} + 17 x - 30$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.6.1

Set $6 x^{2} + 17 x - 30$ equal to $0$ .

$6 x^{2} + 17 x - 30 = 0$

Step 5.6.2

Solve $6 x^{2} + 17 x - 30 = 0$ for $x$ .

Tap for more steps...

Step 5.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.6.2.2

Substitute the values $a = 6$ , $b = 17$ , and $c = - 30$ into the quadratic formula and solve for $x$ .

$\frac{- 17 \pm \sqrt{17^{2} - 4 \cdot (6 \cdot - 30)}}{2 \cdot 6}$

Step 5.6.2.3

Simplify.

Tap for more steps...

Step 5.6.2.3.1

Simplify the numerator.

Tap for more steps...

Step 5.6.2.3.1.1

Raise $17$ to the power of $2$ .

$x = \frac{- 17 \pm \sqrt{289 - 4 \cdot 6 \cdot - 30}}{2 \cdot 6}$

Step 5.6.2.3.1.2

Multiply $- 4 \cdot 6 \cdot - 30$ .

Tap for more steps...

Step 5.6.2.3.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 17 \pm \sqrt{289 - 24 \cdot - 30}}{2 \cdot 6}$

Step 5.6.2.3.1.2.2

Multiply $- 24$ by $- 30$ .

$x = \frac{- 17 \pm \sqrt{289 + 720}}{2 \cdot 6}$

Step 5.6.2.3.1.3

Add $289$ and $720$ .

$x = \frac{- 17 \pm \sqrt{1009}}{2 \cdot 6}$

Step 5.6.2.3.2

Multiply $2$ by $6$ .

$x = \frac{- 17 \pm \sqrt{1009}}{12}$

Step 5.6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 5.6.2.4.1

Simplify the numerator.

Tap for more steps...

Step 5.6.2.4.1.1

Raise $17$ to the power of $2$ .

$x = \frac{- 17 \pm \sqrt{289 - 4 \cdot 6 \cdot - 30}}{2 \cdot 6}$

Step 5.6.2.4.1.2

Multiply $- 4 \cdot 6 \cdot - 30$ .

Tap for more steps...

Step 5.6.2.4.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 17 \pm \sqrt{289 - 24 \cdot - 30}}{2 \cdot 6}$

Step 5.6.2.4.1.2.2

Multiply $- 24$ by $- 30$ .

$x = \frac{- 17 \pm \sqrt{289 + 720}}{2 \cdot 6}$

Step 5.6.2.4.1.3

Add $289$ and $720$ .

$x = \frac{- 17 \pm \sqrt{1009}}{2 \cdot 6}$

Step 5.6.2.4.2

Multiply $2$ by $6$ .

$x = \frac{- 17 \pm \sqrt{1009}}{12}$

Step 5.6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 17 + \sqrt{1009}}{12}$

Step 5.6.2.4.4

Rewrite $- 17$ as $- 1 (17)$ .

$x = \frac{- 1 \cdot 17 + \sqrt{1009}}{12}$

Step 5.6.2.4.5

Factor $- 1$ out of $\sqrt{1009}$ .

$x = \frac{- 1 \cdot 17 - 1 (- \sqrt{1009})}{12}$

Step 5.6.2.4.6

Factor $- 1$ out of $- 1 (17) - 1 (- \sqrt{1009})$ .

$x = \frac{- 1 (17 - \sqrt{1009})}{12}$

Step 5.6.2.4.7

Move the negative in front of the fraction.

$x = - \frac{17 - \sqrt{1009}}{12}$

Step 5.6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 5.6.2.5.1

Simplify the numerator.

Tap for more steps...

Step 5.6.2.5.1.1

Raise $17$ to the power of $2$ .

$x = \frac{- 17 \pm \sqrt{289 - 4 \cdot 6 \cdot - 30}}{2 \cdot 6}$

Step 5.6.2.5.1.2

Multiply $- 4 \cdot 6 \cdot - 30$ .

Tap for more steps...

Step 5.6.2.5.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 17 \pm \sqrt{289 - 24 \cdot - 30}}{2 \cdot 6}$

Step 5.6.2.5.1.2.2

Multiply $- 24$ by $- 30$ .

$x = \frac{- 17 \pm \sqrt{289 + 720}}{2 \cdot 6}$

Step 5.6.2.5.1.3

Add $289$ and $720$ .

$x = \frac{- 17 \pm \sqrt{1009}}{2 \cdot 6}$

Step 5.6.2.5.2

Multiply $2$ by $6$ .

$x = \frac{- 17 \pm \sqrt{1009}}{12}$

Step 5.6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 17 - \sqrt{1009}}{12}$

Step 5.6.2.5.4

Rewrite $- 17$ as $- 1 (17)$ .

$x = \frac{- 1 \cdot 17 - \sqrt{1009}}{12}$

Step 5.6.2.5.5

Factor $- 1$ out of $- \sqrt{1009}$ .

$x = \frac{- 1 \cdot 17 - (\sqrt{1009})}{12}$

Step 5.6.2.5.6

Factor $- 1$ out of $- 1 (17) - (\sqrt{1009})$ .

$x = \frac{- 1 (17 + \sqrt{1009})}{12}$

Step 5.6.2.5.7

Move the negative in front of the fraction.

$x = - \frac{17 + \sqrt{1009}}{12}$

Step 5.6.2.6

The final answer is the combination of both solutions.

$x = - \frac{17 - \sqrt{1009}}{12}, - \frac{17 + \sqrt{1009}}{12}$

Step 5.7

The final solution is all the values that make $x^{2} (x - 2) (6 x^{2} + 17 x - 30) = 0$ true.

$x = 0, 2, - \frac{17 - \sqrt{1009}}{12}, - \frac{17 + \sqrt{1009}}{12}$

Step 6

Find the values where the derivative is undefined.

Tap for more steps...

Step 6.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 7

Critical points to evaluate.

$x = 0, 2, - \frac{17 - \sqrt{1009}}{12}, - \frac{17 + \sqrt{1009}}{12}$

Step 8

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.

$30 {(0)}^{4} + 20 {(0)}^{3} - 192 {(0)}^{2} + 120 (0)$

Step 9

Evaluate the second derivative.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

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

$30 \cdot 0 + 20 {(0)}^{3} - 192 {(0)}^{2} + 120 (0)$

Step 9.1.2

Multiply $30$ by $0$ .

$0 + 20 {(0)}^{3} - 192 {(0)}^{2} + 120 (0)$

Step 9.1.3

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

$0 + 20 \cdot 0 - 192 {(0)}^{2} + 120 (0)$

Step 9.1.4

Multiply $20$ by $0$ .

$0 + 0 - 192 {(0)}^{2} + 120 (0)$

Step 9.1.5

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

$0 + 0 - 192 \cdot 0 + 120 (0)$

Step 9.1.6

Multiply $- 192$ by $0$ .

$0 + 0 + 0 + 120 (0)$

Step 9.1.7

Multiply $120$ by $0$ .

$0 + 0 + 0 + 0$

Step 9.2

Simplify by adding numbers.

Tap for more steps...

Step 9.2.1

Add $0$ and $0$ .

$0 + 0 + 0$

Step 9.2.2

Add $0$ and $0$ .

$0 + 0$

Step 9.2.3

Add $0$ and $0$ .

$0$

Step 10

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

Tap for more steps...

Step 10.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - \frac{17 + \sqrt{1009}}{12}) \cup (- \frac{17 + \sqrt{1009}}{12}, 0) \cup (0, - \frac{17 - \sqrt{1009}}{12}) \cup (- \frac{17 - \sqrt{1009}}{12}, 2) \cup (2, \infty)$

Step 10.2

Substitute any number, such as $- 7$ , from the interval $(- \infty, - \frac{17 + \sqrt{1009}}{12})$ in the first derivative $6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.2.1

Replace the variable $x$ with $- 7$ in the expression.

$f' (- 7) = 6 {(- 7)}^{5} + 5 {(- 7)}^{4} - 64 {(- 7)}^{3} + 60 {(- 7)}^{2}$

Step 10.2.2

Simplify the result.

Tap for more steps...

Step 10.2.2.1

Simplify each term.

Tap for more steps...

Step 10.2.2.1.1

Raise $- 7$ to the power of $5$ .

$f' (- 7) = 6 \cdot - 16807 + 5 {(- 7)}^{4} - 64 {(- 7)}^{3} + 60 {(- 7)}^{2}$

Step 10.2.2.1.2

Multiply $6$ by $- 16807$ .

$f' (- 7) = - 100842 + 5 {(- 7)}^{4} - 64 {(- 7)}^{3} + 60 {(- 7)}^{2}$

Step 10.2.2.1.3

Raise $- 7$ to the power of $4$ .

$f' (- 7) = - 100842 + 5 \cdot 2401 - 64 {(- 7)}^{3} + 60 {(- 7)}^{2}$

Step 10.2.2.1.4

Multiply $5$ by $2401$ .

$f' (- 7) = - 100842 + 12005 - 64 {(- 7)}^{3} + 60 {(- 7)}^{2}$

Step 10.2.2.1.5

Raise $- 7$ to the power of $3$ .

$f' (- 7) = - 100842 + 12005 - 64 \cdot - 343 + 60 {(- 7)}^{2}$

Step 10.2.2.1.6

Multiply $- 64$ by $- 343$ .

$f' (- 7) = - 100842 + 12005 + 21952 + 60 {(- 7)}^{2}$

Step 10.2.2.1.7

Raise $- 7$ to the power of $2$ .

$f' (- 7) = - 100842 + 12005 + 21952 + 60 \cdot 49$

Step 10.2.2.1.8

Multiply $60$ by $49$ .

$f' (- 7) = - 100842 + 12005 + 21952 + 2940$

Step 10.2.2.2

Simplify by adding numbers.

Tap for more steps...

Step 10.2.2.2.1

Add $- 100842$ and $12005$ .

$f' (- 7) = - 88837 + 21952 + 2940$

Step 10.2.2.2.2

Add $- 88837$ and $21952$ .

$f' (- 7) = - 66885 + 2940$

Step 10.2.2.2.3

Add $- 66885$ and $2940$ .

$f' (- 7) = - 63945$

Step 10.2.2.3

The final answer is $- 63945$ .

$- 63945$

Step 10.3

Substitute any number, such as $- 2$ , from the interval $(- \frac{17 + \sqrt{1009}}{12}, 0)$ in the first derivative $6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.3.1

Replace the variable $x$ with $- 2$ in the expression.

$f' (- 2) = 6 {(- 2)}^{5} + 5 {(- 2)}^{4} - 64 {(- 2)}^{3} + 60 {(- 2)}^{2}$

Step 10.3.2

Simplify the result.

Tap for more steps...

Step 10.3.2.1

Simplify each term.

Tap for more steps...

Step 10.3.2.1.1

Raise $- 2$ to the power of $5$ .

$f' (- 2) = 6 \cdot - 32 + 5 {(- 2)}^{4} - 64 {(- 2)}^{3} + 60 {(- 2)}^{2}$

Step 10.3.2.1.2

Multiply $6$ by $- 32$ .

$f' (- 2) = - 192 + 5 {(- 2)}^{4} - 64 {(- 2)}^{3} + 60 {(- 2)}^{2}$

Step 10.3.2.1.3

Raise $- 2$ to the power of $4$ .

$f' (- 2) = - 192 + 5 \cdot 16 - 64 {(- 2)}^{3} + 60 {(- 2)}^{2}$

Step 10.3.2.1.4

Multiply $5$ by $16$ .

$f' (- 2) = - 192 + 80 - 64 {(- 2)}^{3} + 60 {(- 2)}^{2}$

Step 10.3.2.1.5

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

$f' (- 2) = - 192 + 80 - 64 \cdot - 8 + 60 {(- 2)}^{2}$

Step 10.3.2.1.6

Multiply $- 64$ by $- 8$ .

$f' (- 2) = - 192 + 80 + 512 + 60 {(- 2)}^{2}$

Step 10.3.2.1.7

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

$f' (- 2) = - 192 + 80 + 512 + 60 \cdot 4$

Step 10.3.2.1.8

Multiply $60$ by $4$ .

$f' (- 2) = - 192 + 80 + 512 + 240$

Step 10.3.2.2

Simplify by adding numbers.

Tap for more steps...

Step 10.3.2.2.1

Add $- 192$ and $80$ .

$f' (- 2) = - 112 + 512 + 240$

Step 10.3.2.2.2

Add $- 112$ and $512$ .

$f' (- 2) = 400 + 240$

Step 10.3.2.2.3

Add $400$ and $240$ .

$f' (- 2) = 640$

Step 10.3.2.3

The final answer is $640$ .

$640$

Step 10.4

Substitute any number, such as $1$ , from the interval $(0, - \frac{17 - \sqrt{1009}}{12})$ in the first derivative $6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.4.1

Replace the variable $x$ with $1$ in the expression.

$f' (1) = 6 {(1)}^{5} + 5 {(1)}^{4} - 64 {(1)}^{3} + 60 {(1)}^{2}$

Step 10.4.2

Simplify the result.

Tap for more steps...

Step 10.4.2.1

Simplify each term.

Tap for more steps...

Step 10.4.2.1.1

One to any power is one.

$f' (1) = 6 \cdot 1 + 5 {(1)}^{4} - 64 {(1)}^{3} + 60 {(1)}^{2}$

Step 10.4.2.1.2

Multiply $6$ by $1$ .

$f' (1) = 6 + 5 {(1)}^{4} - 64 {(1)}^{3} + 60 {(1)}^{2}$

Step 10.4.2.1.3

One to any power is one.

$f' (1) = 6 + 5 \cdot 1 - 64 {(1)}^{3} + 60 {(1)}^{2}$

Step 10.4.2.1.4

Multiply $5$ by $1$ .

$f' (1) = 6 + 5 - 64 {(1)}^{3} + 60 {(1)}^{2}$

Step 10.4.2.1.5

One to any power is one.

$f' (1) = 6 + 5 - 64 \cdot 1 + 60 {(1)}^{2}$

Step 10.4.2.1.6

Multiply $- 64$ by $1$ .

$f' (1) = 6 + 5 - 64 + 60 {(1)}^{2}$

Step 10.4.2.1.7

One to any power is one.

$f' (1) = 6 + 5 - 64 + 60 \cdot 1$

Step 10.4.2.1.8

Multiply $60$ by $1$ .

$f' (1) = 6 + 5 - 64 + 60$

Step 10.4.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 10.4.2.2.1

Add $6$ and $5$ .

$f' (1) = 11 - 64 + 60$

Step 10.4.2.2.2

Subtract $64$ from $11$ .

$f' (1) = - 53 + 60$

Step 10.4.2.2.3

Add $- 53$ and $60$ .

$f' (1) = 7$

Step 10.4.2.3

The final answer is $7$ .

$7$

Step 10.5

Substitute any number, such as $1.62$ , from the interval $(- \frac{17 - \sqrt{1009}}{12}, 2)$ in the first derivative $6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.5.1

Replace the variable $x$ with $1.62$ in the expression.

$f' (1.62) = 6 {(1.62)}^{5} + 5 {(1.62)}^{4} - 64 {(1.62)}^{3} + 60 {(1.62)}^{2}$

Step 10.5.2

Simplify the result.

Tap for more steps...

Step 10.5.2.1

Simplify each term.

Tap for more steps...

Step 10.5.2.1.1

Raise $1.62$ to the power of $5$ .

$f' (1.62) = 6 \cdot 11.15771008 + 5 {(1.62)}^{4} - 64 {(1.62)}^{3} + 60 {(1.62)}^{2}$

Step 10.5.2.1.2

Multiply $6$ by $11.15771008$ .

$f' (1.62) = 66.94626049 + 5 {(1.62)}^{4} - 64 {(1.62)}^{3} + 60 {(1.62)}^{2}$

Step 10.5.2.1.3

Raise $1.62$ to the power of $4$ .

$f' (1.62) = 66.94626049 + 5 \cdot 6.88747536 - 64 {(1.62)}^{3} + 60 {(1.62)}^{2}$

Step 10.5.2.1.4

Multiply $5$ by $6.88747536$ .

$f' (1.62) = 66.94626049 + 34.4373768 - 64 {(1.62)}^{3} + 60 {(1.62)}^{2}$

Step 10.5.2.1.5

Raise $1.62$ to the power of $3$ .

$f' (1.62) = 66.94626049 + 34.4373768 - 64 \cdot 4.251528 + 60 {(1.62)}^{2}$

Step 10.5.2.1.6

Multiply $- 64$ by $4.251528$ .

$f' (1.62) = 66.94626049 + 34.4373768 - 272.097792 + 60 {(1.62)}^{2}$

Step 10.5.2.1.7

Raise $1.62$ to the power of $2$ .

$f' (1.62) = 66.94626049 + 34.4373768 - 272.097792 + 60 \cdot 2.6244$

Step 10.5.2.1.8

Multiply $60$ by $2.6244$ .

$f' (1.62) = 66.94626049 + 34.4373768 - 272.097792 + 157.464$

Step 10.5.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 10.5.2.2.1

Add $66.94626049$ and $34.4373768$ .

$f' (1.62) = 101.38363729 - 272.097792 + 157.464$

Step 10.5.2.2.2

Subtract $272.097792$ from $101.38363729$ .

$f' (1.62) = - 170.7141547 + 157.464$

Step 10.5.2.2.3

Add $- 170.7141547$ and $157.464$ .

$f' (1.62) = - 13.2501547$

Step 10.5.2.3

The final answer is $- 13.2501547$ .

$- 13.2501547$

Step 10.6

Substitute any number, such as $4$ , from the interval $(2, \infty)$ in the first derivative $6 x^{5} + 5 x^{4} - 64 x^{3} + 60 x^{2}$ to check if the result is negative or positive.

Tap for more steps...

Step 10.6.1

Replace the variable $x$ with $4$ in the expression.

$f' (4) = 6 {(4)}^{5} + 5 {(4)}^{4} - 64 {(4)}^{3} + 60 {(4)}^{2}$

Step 10.6.2

Simplify the result.

Tap for more steps...

Step 10.6.2.1

Simplify each term.

Tap for more steps...

Step 10.6.2.1.1

Raise $4$ to the power of $5$ .

$f' (4) = 6 \cdot 1024 + 5 {(4)}^{4} - 64 {(4)}^{3} + 60 {(4)}^{2}$

Step 10.6.2.1.2

Multiply $6$ by $1024$ .

$f' (4) = 6144 + 5 {(4)}^{4} - 64 {(4)}^{3} + 60 {(4)}^{2}$

Step 10.6.2.1.3

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

$f' (4) = 6144 + 5 \cdot 256 - 64 {(4)}^{3} + 60 {(4)}^{2}$

Step 10.6.2.1.4

Multiply $5$ by $256$ .

$f' (4) = 6144 + 1280 - 64 {(4)}^{3} + 60 {(4)}^{2}$

Step 10.6.2.1.5

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

$f' (4) = 6144 + 1280 - 64 \cdot 64 + 60 {(4)}^{2}$

Step 10.6.2.1.6

Multiply $- 64$ by $64$ .

$f' (4) = 6144 + 1280 - 4096 + 60 {(4)}^{2}$

Step 10.6.2.1.7

Raise $4$ to the power of $2$ .

$f' (4) = 6144 + 1280 - 4096 + 60 \cdot 16$

Step 10.6.2.1.8

Multiply $60$ by $16$ .

$f' (4) = 6144 + 1280 - 4096 + 960$

Step 10.6.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 10.6.2.2.1

Add $6144$ and $1280$ .

$f' (4) = 7424 - 4096 + 960$

Step 10.6.2.2.2

Subtract $4096$ from $7424$ .

$f' (4) = 3328 + 960$

Step 10.6.2.2.3

Add $3328$ and $960$ .

$f' (4) = 4288$

Step 10.6.2.3

The final answer is $4288$ .

$4288$

Step 10.7

Since the first derivative changed signs from negative to positive around $x = - \frac{17 + \sqrt{1009}}{12}$ , then $x = - \frac{17 + \sqrt{1009}}{12}$ is a local minimum.

$x = - \frac{17 + \sqrt{1009}}{12}$ is a local minimum

Step 10.8

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 10.9

Since the first derivative changed signs from positive to negative around $x = - \frac{17 - \sqrt{1009}}{12}$ , then $x = - \frac{17 - \sqrt{1009}}{12}$ is a local maximum.

$x = - \frac{17 - \sqrt{1009}}{12}$ is a local maximum

Step 10.10

Since the first derivative changed signs from negative to positive around $x = 2$ , then $x = 2$ is a local minimum.

$x = 2$ is a local minimum

Step 10.11

These are the local extrema for $f (x) = x^{3} {(x - 2)}^{2} (x + 5)$ .

$x = - \frac{17 + \sqrt{1009}}{12}$ is a local minimum

$x = - \frac{17 - \sqrt{1009}}{12}$ is a local maximum

$x = 2$ is a local minimum

$x = - \frac{17 + \sqrt{1009}}{12}$ is a local minimum

$x = - \frac{17 - \sqrt{1009}}{12}$ is a local maximum

$x = 2$ is a local minimum

Step 11