Calculus Examples

$\frac{x^{3}}{x^{2} - 9}$

Step 1

Write $\frac{x^{3}}{x^{2} - 9}$ as a function.

$f (x) = \frac{x^{3}}{x^{2} - 9}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 $f (x) = x^{3}$ and $g (x) = x^{2} - 9$ .

$f' (x) = \frac{(x^{2} - 9) \frac{d}{d x} (x^{3}) - x^{3} \frac{d}{d x} (x^{2} - 9)}{{(x^{2} - 9)}^{2}}$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

$f' (x) = \frac{(x^{2} - 9) (3 x^{2}) - x^{3} \frac{d}{d x} (x^{2} - 9)}{{(x^{2} - 9)}^{2}}$

Step 2.1.2.2

Move $3$ to the left of $x^{2} - 9$ .

$f' (x) = \frac{3 \cdot ((x^{2} - 9) x^{2}) - x^{3} \frac{d}{d x} (x^{2} - 9)}{{(x^{2} - 9)}^{2}}$

Step 2.1.2.3

By the Sum Rule, the derivative of $x^{2} - 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9]$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - x^{3} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 9))}{{(x^{2} - 9)}^{2}}$

Step 2.1.2.4

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

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - x^{3} (2 x + \frac{d}{d x} (- 9))}{{(x^{2} - 9)}^{2}}$

Step 2.1.2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - x^{3} (2 x + 0)}{{(x^{2} - 9)}^{2}}$

Step 2.1.2.6

Simplify the expression.

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Step 2.1.2.6.1

Add $2 x$ and $0$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - x^{3} (2 x)}{{(x^{2} - 9)}^{2}}$

Step 2.1.2.6.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - 2 x^{3} x}{{(x^{2} - 9)}^{2}}$

Step 2.1.3

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

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - 2 (x \cdot x^{3})}{{(x^{2} - 9)}^{2}}$

Step 2.1.4

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

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - 2 x^{1 + 3}}{{(x^{2} - 9)}^{2}}$

Step 2.1.5

Add $1$ and $3$ .

$f' (x) = \frac{3 (x^{2} - 9) x^{2} - 2 x^{4}}{{(x^{2} - 9)}^{2}}$

Step 2.1.6

Simplify.

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Step 2.1.6.1

Apply the distributive property.

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

Step 2.1.6.2

Apply the distributive property.

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

Step 2.1.6.3

Simplify the numerator.

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Step 2.1.6.3.1

Simplify each term.

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Step 2.1.6.3.1.1

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

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Step 2.1.6.3.1.1.1

Move $x^{2}$ .

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

Step 2.1.6.3.1.1.2

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

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

Step 2.1.6.3.1.1.3

Add $2$ and $2$ .

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

Step 2.1.6.3.1.2

Multiply $3$ by $- 9$ .

$f' (x) = \frac{3 x^{4} - 27 x^{2} - 2 x^{4}}{{(x^{2} - 9)}^{2}}$

Step 2.1.6.3.2

Subtract $2 x^{4}$ from $3 x^{4}$ .

$f' (x) = \frac{x^{4} - 27 x^{2}}{{(x^{2} - 9)}^{2}}$

Step 2.1.6.4

Factor $x^{2}$ out of $x^{4} - 27 x^{2}$ .

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Step 2.1.6.4.1

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

$f' (x) = \frac{x^{2} x^{2} - 27 x^{2}}{{(x^{2} - 9)}^{2}}$

Step 2.1.6.4.2

Factor $x^{2}$ out of $- 27 x^{2}$ .

$f' (x) = \frac{x^{2} x^{2} + x^{2} \cdot - 27}{{(x^{2} - 9)}^{2}}$

Step 2.1.6.4.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 27$ .

$f' (x) = \frac{x^{2} (x^{2} - 27)}{{(x^{2} - 9)}^{2}}$

Step 2.1.6.5

Simplify the denominator.

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Step 2.1.6.5.1

Rewrite $9$ as $3^{2}$ .

$f' (x) = \frac{x^{2} (x^{2} - 27)}{{(x^{2} - 3^{2})}^{2}}$

Step 2.1.6.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$f' (x) = \frac{x^{2} (x^{2} - 27)}{{((x + 3) (x - 3))}^{2}}$

Step 2.1.6.5.3

Apply the product rule to $(x + 3) (x - 3)$ .

$f' (x) = \frac{x^{2} (x^{2} - 27)}{{(x + 3)}^{2} {(x - 3)}^{2}}$

Step 2.2

Find the second derivative.

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Step 2.2.1

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} \frac{d}{d x} (x^{2} (x^{2} - 27)) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.2

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 $f (x) = x^{2}$ and $g (x) = x^{2} - 27$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (x^{2} \frac{d}{d x} (x^{2} - 27) + (x^{2} - 27) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.3

Differentiate.

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Step 2.2.3.1

By the Sum Rule, the derivative of $x^{2} - 27$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 27]$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 27)) + (x^{2} - 27) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.3.2

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (x^{2} (2 x + \frac{d}{d x} (- 27)) + (x^{2} - 27) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.3.3

Since $- 27$ is constant with respect to $x$ , the derivative of $- 27$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (x^{2} (2 x + 0) + (x^{2} - 27) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.3.4

Add $2 x$ and $0$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (x^{2} (2 x) + (x^{2} - 27) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.4

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

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Step 2.2.4.1

Move $x$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (x \cdot x^{2} \cdot 2 + (x^{2} - 27) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.4.2

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

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Step 2.2.4.2.1

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

Step 2.2.4.2.2

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (x^{1 + 2} \cdot 2 + (x^{2} - 27) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.4.3

Add $1$ and $2$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (x^{3} \cdot 2 + (x^{2} - 27) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.5

Differentiate using the Power Rule.

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Step 2.2.5.1

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 \cdot x^{3} + (x^{2} - 27) \frac{d}{d x} (x^{2})) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.5.2

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + (x^{2} - 27) (2 x)) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.5.3

Move $2$ to the left of $x^{2} - 27$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 \cdot ((x^{2} - 27) x)) - x^{2} (x^{2} - 27) \frac{d}{d x} ({(x + 3)}^{2} {(x - 3)}^{2})}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.6

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) ({(x + 3)}^{2} \frac{d}{d x} ({(x - 3)}^{2}) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x - 3$ .

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Step 2.2.7.1

To apply the Chain Rule, set $u_{1}$ as $x - 3$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) ({(x + 3)}^{2} (\frac{d}{d u (1)} ({u_{1}}^{2}) \frac{d}{d x} (x - 3)) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.7.2

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) ({(x + 3)}^{2} (2 u_{1} \frac{d}{d x} (x - 3)) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.7.3

Replace all occurrences of $u_{1}$ with $x - 3$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) ({(x + 3)}^{2} (2 (x - 3) \frac{d}{d x} (x - 3)) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.8

Differentiate.

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Step 2.2.8.1

Move $2$ to the left of ${(x + 3)}^{2}$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 \cdot ({(x + 3)}^{2} (x - 3)) \frac{d}{d x} (x - 3) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.8.2

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) (\frac{d}{d x} (x) + \frac{d}{d x} (- 3)) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.8.3

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) (1 + \frac{d}{d x} (- 3)) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.8.4

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) (1 + 0) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.8.5

Simplify the expression.

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Step 2.2.8.5.1

Add $1$ and $0$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) \cdot 1 + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.8.5.2

Multiply $2$ by $1$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + {(x - 3)}^{2} \frac{d}{d x} ({(x + 3)}^{2}))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x + 3$ .

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Step 2.2.9.1

To apply the Chain Rule, set $u_{2}$ as $x + 3$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + {(x - 3)}^{2} (\frac{d}{d u (2)} ({u_{2}}^{2}) \frac{d}{d x} (x + 3)))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.9.2

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + {(x - 3)}^{2} (2 u_{2} \frac{d}{d x} (x + 3)))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.9.3

Replace all occurrences of $u_{2}$ with $x + 3$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + {(x - 3)}^{2} (2 (x + 3) \frac{d}{d x} (x + 3)))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.10

Differentiate.

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Step 2.2.10.1

Move $2$ to the left of ${(x - 3)}^{2}$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + 2 \cdot ({(x - 3)}^{2} (x + 3)) \frac{d}{d x} (x + 3))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.10.2

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3) (\frac{d}{d x} (x) + \frac{d}{d x} (3)))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.10.3

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3) (1 + \frac{d}{d x} (3)))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.10.4

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

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3) (1 + 0))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.10.5

Simplify the expression.

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Step 2.2.10.5.1

Add $1$ and $0$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3) \cdot 1)}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.10.5.2

Multiply $2$ by $1$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2} {(x - 3)}^{2})}^{2}}$

Step 2.2.11

Simplify.

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Step 2.2.11.1

Apply the product rule to ${(x + 3)}^{2} {(x - 3)}^{2}$ .

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 (x^{2} - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.2

Apply the distributive property.

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + (2 x^{2} + 2 \cdot - 27) x) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.3

Apply the distributive property.

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) - x^{2} (x^{2} - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.4

Apply the distributive property.

$f'' (x) = \frac{{(x + 3)}^{2} {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5

Simplify the numerator.

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Step 2.2.11.5.1

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

$f'' (x) = \frac{(x + 3) (x + 3) {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.2

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

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Step 2.2.11.5.2.1

Apply the distributive property.

$f'' (x) = \frac{(x (x + 3) + 3 (x + 3)) {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.2.2

Apply the distributive property.

$f'' (x) = \frac{(x \cdot x + x \cdot 3 + 3 (x + 3)) {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.2.3

Apply the distributive property.

$f'' (x) = \frac{(x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.3

Simplify and combine like terms.

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Step 2.2.11.5.3.1

Simplify each term.

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Step 2.2.11.5.3.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{(x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.3.1.2

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

$f'' (x) = \frac{(x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.3.1.3

Multiply $3$ by $3$ .

$f'' (x) = \frac{(x^{2} + 3 x + 3 x + 9) {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.3.2

Add $3 x$ and $3 x$ .

$f'' (x) = \frac{(x^{2} + 6 x + 9) {(x - 3)}^{2} (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.4

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

$f'' (x) = \frac{(x^{2} + 6 x + 9) ((x - 3) (x - 3)) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.5

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

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Step 2.2.11.5.5.1

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 6 x + 9) (x (x - 3) - 3 (x - 3)) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.5.2

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 6 x + 9) (x \cdot x + x \cdot - 3 - 3 (x - 3)) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.5.3

Apply the distributive property.

$f'' (x) = \frac{(x^{2} + 6 x + 9) (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.6

Simplify and combine like terms.

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Step 2.2.11.5.6.1

Simplify each term.

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Step 2.2.11.5.6.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{(x^{2} + 6 x + 9) (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.6.1.2

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

$f'' (x) = \frac{(x^{2} + 6 x + 9) (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.6.1.3

Multiply $- 3$ by $- 3$ .

$f'' (x) = \frac{(x^{2} + 6 x + 9) (x^{2} - 3 x - 3 x + 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.6.2

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

$f'' (x) = \frac{(x^{2} + 6 x + 9) (x^{2} - 6 x + 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.7

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

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} (- 6 x) + x^{2} \cdot 9 + 6 x \cdot x^{2} + 6 x (- 6 x) + 6 x \cdot 9 + 9 x^{2} + 9 (- 6 x) + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.8

Combine the opposite terms in $x^{2} x^{2} + x^{2} (- 6 x) + x^{2} \cdot 9 + 6 x \cdot x^{2} + 6 x (- 6 x) + 6 x \cdot 9 + 9 x^{2} + 9 (- 6 x) + 9 \cdot 9$ .

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Step 2.2.11.5.8.1

Reorder the factors in the terms $x^{2} (- 6 x)$ and $6 x \cdot x^{2}$ .

$f'' (x) = \frac{(x^{2} x^{2} - 6 x^{2} x + x^{2} \cdot 9 + 6 x^{2} x + 6 x (- 6 x) + 6 x \cdot 9 + 9 x^{2} + 9 (- 6 x) + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.8.2

Add $- 6 x^{2} x$ and $6 x^{2} x$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 9 + 0 + 6 x (- 6 x) + 6 x \cdot 9 + 9 x^{2} + 9 (- 6 x) + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.8.3

Add $x^{2} x^{2} + x^{2} \cdot 9$ and $0$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 9 + 6 x (- 6 x) + 6 x \cdot 9 + 9 x^{2} + 9 (- 6 x) + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.8.4

Reorder the factors in the terms $6 x \cdot 9$ and $9 (- 6 x)$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 9 + 6 x (- 6 x) + 6 \cdot (9 x) + 9 x^{2} - 6 \cdot (9 x) + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.8.5

Subtract $6 \cdot 9 x$ from $6 \cdot 9 x$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 9 + 6 x (- 6 x) + 9 x^{2} + 0 + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.8.6

Add $x^{2} x^{2} + x^{2} \cdot 9 + 6 x (- 6 x) + 9 x^{2}$ and $0$ .

$f'' (x) = \frac{(x^{2} x^{2} + x^{2} \cdot 9 + 6 x (- 6 x) + 9 x^{2} + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.9

Simplify each term.

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Step 2.2.11.5.9.1

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

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Step 2.2.11.5.9.1.1

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

$f'' (x) = \frac{(x^{2 + 2} + x^{2} \cdot 9 + 6 x (- 6 x) + 9 x^{2} + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.9.1.2

Add $2$ and $2$ .

$f'' (x) = \frac{(x^{4} + x^{2} \cdot 9 + 6 x (- 6 x) + 9 x^{2} + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.9.2

Move $9$ to the left of $x^{2}$ .

$f'' (x) = \frac{(x^{4} + 9 \cdot x^{2} + 6 x (- 6 x) + 9 x^{2} + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.9.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{(x^{4} + 9 x^{2} + 6 \cdot (- 6 x \cdot x) + 9 x^{2} + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.9.4

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

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Step 2.2.11.5.9.4.1

Move $x$ .

$f'' (x) = \frac{(x^{4} + 9 x^{2} + 6 \cdot (- 6 (x \cdot x)) + 9 x^{2} + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.9.4.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{(x^{4} + 9 x^{2} + 6 \cdot (- 6 x^{2}) + 9 x^{2} + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.9.5

Multiply $6$ by $- 6$ .

$f'' (x) = \frac{(x^{4} + 9 x^{2} - 36 x^{2} + 9 x^{2} + 9 \cdot 9) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.9.6

Multiply $9$ by $9$ .

$f'' (x) = \frac{(x^{4} + 9 x^{2} - 36 x^{2} + 9 x^{2} + 81) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.10

Subtract $36 x^{2}$ from $9 x^{2}$ .

$f'' (x) = \frac{(x^{4} - 27 x^{2} + 9 x^{2} + 81) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.11

Add $- 27 x^{2}$ and $9 x^{2}$ .

$f'' (x) = \frac{(x^{4} - 18 x^{2} + 81) (2 x^{3} + 2 x^{2} x + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.12

Simplify each term.

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Step 2.2.11.5.12.1

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

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Step 2.2.11.5.12.1.1

Move $x$ .

$f'' (x) = \frac{(x^{4} - 18 x^{2} + 81) (2 x^{3} + 2 (x \cdot x^{2}) + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.12.1.2

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

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Step 2.2.11.5.12.1.2.1

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

Step 2.2.11.5.12.1.2.2

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

$f'' (x) = \frac{(x^{4} - 18 x^{2} + 81) (2 x^{3} + 2 x^{1 + 2} + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.12.1.3

Add $1$ and $2$ .

$f'' (x) = \frac{(x^{4} - 18 x^{2} + 81) (2 x^{3} + 2 x^{3} + 2 \cdot (- 27 x)) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.12.2

Multiply $2$ by $- 27$ .

$f'' (x) = \frac{(x^{4} - 18 x^{2} + 81) (2 x^{3} + 2 x^{3} - 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.13

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

$f'' (x) = \frac{(x^{4} - 18 x^{2} + 81) (4 x^{3} - 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.14

Expand $(x^{4} - 18 x^{2} + 81) (4 x^{3} - 54 x)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{x^{4} (4 x^{3}) + x^{4} (- 54 x) - 18 x^{2} (4 x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15

Simplify each term.

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Step 2.2.11.5.15.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{4} x^{3} + x^{4} (- 54 x) - 18 x^{2} (4 x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 2.2.11.5.15.2.1

Move $x^{3}$ .

$f'' (x) = \frac{4 (x^{3} x^{4}) + x^{4} (- 54 x) - 18 x^{2} (4 x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.2.2

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

$f'' (x) = \frac{4 x^{3 + 4} + x^{4} (- 54 x) - 18 x^{2} (4 x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.2.3

Add $3$ and $4$ .

$f'' (x) = \frac{4 x^{7} + x^{4} (- 54 x) - 18 x^{2} (4 x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 54 x^{4} x - 18 x^{2} (4 x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.4

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

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Step 2.2.11.5.15.4.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 54 (x \cdot x^{4}) - 18 x^{2} (4 x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.4.2

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

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Step 2.2.11.5.15.4.2.1

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

Step 2.2.11.5.15.4.2.2

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

$f'' (x) = \frac{4 x^{7} - 54 x^{1 + 4} - 18 x^{2} (4 x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.4.3

Add $1$ and $4$ .

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 18 x^{2} (4 x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 18 \cdot (4 x^{2} x^{3}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.6

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

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Step 2.2.11.5.15.6.1

Move $x^{3}$ .

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 18 \cdot (4 (x^{3} x^{2})) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.6.2

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

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 18 \cdot (4 x^{3 + 2}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.6.3

Add $3$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 18 \cdot (4 x^{5}) - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.7

Multiply $- 18$ by $4$ .

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 72 x^{5} - 18 x^{2} (- 54 x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.8

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 72 x^{5} - 18 \cdot (- 54 x^{2} x) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.9

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

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Step 2.2.11.5.15.9.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 72 x^{5} - 18 \cdot (- 54 (x \cdot x^{2})) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.9.2

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

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Step 2.2.11.5.15.9.2.1

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

Step 2.2.11.5.15.9.2.2

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

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 72 x^{5} - 18 \cdot (- 54 x^{1 + 2}) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.9.3

Add $1$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 72 x^{5} - 18 \cdot (- 54 x^{3}) + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.10

Multiply $- 18$ by $- 54$ .

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 72 x^{5} + 972 x^{3} + 81 (4 x^{3}) + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.11

Multiply $4$ by $81$ .

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 72 x^{5} + 972 x^{3} + 324 x^{3} + 81 (- 54 x) + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.15.12

Multiply $- 54$ by $81$ .

$f'' (x) = \frac{4 x^{7} - 54 x^{5} - 72 x^{5} + 972 x^{3} + 324 x^{3} - 4374 x + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.16

Subtract $72 x^{5}$ from $- 54 x^{5}$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 972 x^{3} + 324 x^{3} - 4374 x + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.17

Add $972 x^{3}$ and $324 x^{3}$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{2} x^{2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.18

Simplify each term.

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Step 2.2.11.5.18.1

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

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Step 2.2.11.5.18.1.1

Move $x^{2}$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- (x^{2} x^{2}) - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.18.1.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{2 + 2} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.18.1.3

Add $2$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} - x^{2} \cdot - 27) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.18.2

Multiply $- 27$ by $- 1$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 {(x + 3)}^{2} (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19

Simplify each term.

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Step 2.2.11.5.19.1

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 ((x + 3) (x + 3)) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.2

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

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Step 2.2.11.5.19.2.1

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 (x (x + 3) + 3 (x + 3)) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.2.2

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 (x \cdot x + x \cdot 3 + 3 (x + 3)) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.2.3

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.3

Simplify and combine like terms.

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Step 2.2.11.5.19.3.1

Simplify each term.

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Step 2.2.11.5.19.3.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.3.1.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.3.1.3

Multiply $3$ by $3$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 (x^{2} + 3 x + 3 x + 9) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.3.2

Add $3 x$ and $3 x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 (x^{2} + 6 x + 9) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.4

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) ((2 x^{2} + 2 (6 x) + 2 \cdot 9) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.5

Simplify.

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Step 2.2.11.5.19.5.1

Multiply $6$ by $2$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) ((2 x^{2} + 12 x + 2 \cdot 9) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.5.2

Multiply $2$ by $9$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) ((2 x^{2} + 12 x + 18) (x - 3) + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.6

Expand $(2 x^{2} + 12 x + 18) (x - 3)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{2} x + 2 x^{2} \cdot - 3 + 12 x \cdot x + 12 x \cdot - 3 + 18 x + 18 \cdot - 3 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.7

Simplify each term.

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Step 2.2.11.5.19.7.1

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

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Step 2.2.11.5.19.7.1.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 (x \cdot x^{2}) + 2 x^{2} \cdot - 3 + 12 x \cdot x + 12 x \cdot - 3 + 18 x + 18 \cdot - 3 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.7.1.2

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

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Step 2.2.11.5.19.7.1.2.1

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

Step 2.2.11.5.19.7.1.2.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{1 + 2} + 2 x^{2} \cdot - 3 + 12 x \cdot x + 12 x \cdot - 3 + 18 x + 18 \cdot - 3 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.7.1.3

Add $1$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 2 x^{2} \cdot - 3 + 12 x \cdot x + 12 x \cdot - 3 + 18 x + 18 \cdot - 3 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.7.2

Multiply $- 3$ by $2$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} - 6 x^{2} + 12 x \cdot x + 12 x \cdot - 3 + 18 x + 18 \cdot - 3 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.7.3

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

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Step 2.2.11.5.19.7.3.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} - 6 x^{2} + 12 (x \cdot x) + 12 x \cdot - 3 + 18 x + 18 \cdot - 3 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.7.3.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} - 6 x^{2} + 12 x^{2} + 12 x \cdot - 3 + 18 x + 18 \cdot - 3 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.7.4

Multiply $- 3$ by $12$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} - 6 x^{2} + 12 x^{2} - 36 x + 18 x + 18 \cdot - 3 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.7.5

Multiply $18$ by $- 3$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} - 6 x^{2} + 12 x^{2} - 36 x + 18 x - 54 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.8

Add $- 6 x^{2}$ and $12 x^{2}$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 36 x + 18 x - 54 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.9

Add $- 36 x$ and $18 x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 {(x - 3)}^{2} (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.10

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 ((x - 3) (x - 3)) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.11

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

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Step 2.2.11.5.19.11.1

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 (x (x - 3) - 3 (x - 3)) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.11.2

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 (x \cdot x + x \cdot - 3 - 3 (x - 3)) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.11.3

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.12

Simplify and combine like terms.

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Step 2.2.11.5.19.12.1

Simplify each term.

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Step 2.2.11.5.19.12.1.1

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.12.1.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.12.1.3

Multiply $- 3$ by $- 3$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 (x^{2} - 3 x - 3 x + 9) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.12.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 (x^{2} - 6 x + 9) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.13

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + (2 x^{2} + 2 (- 6 x) + 2 \cdot 9) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.14

Simplify.

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Step 2.2.11.5.19.14.1

Multiply $- 6$ by $2$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + (2 x^{2} - 12 x + 2 \cdot 9) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.14.2

Multiply $2$ by $9$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + (2 x^{2} - 12 x + 18) (x + 3))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.15

Expand $(2 x^{2} - 12 x + 18) (x + 3)$ by multiplying each term in the first expression by each term in the second expression.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{2} x + 2 x^{2} \cdot 3 - 12 x \cdot x - 12 x \cdot 3 + 18 x + 18 \cdot 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.16

Simplify each term.

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Step 2.2.11.5.19.16.1

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

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Step 2.2.11.5.19.16.1.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 (x \cdot x^{2}) + 2 x^{2} \cdot 3 - 12 x \cdot x - 12 x \cdot 3 + 18 x + 18 \cdot 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.16.1.2

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

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Step 2.2.11.5.19.16.1.2.1

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

Step 2.2.11.5.19.16.1.2.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{1 + 2} + 2 x^{2} \cdot 3 - 12 x \cdot x - 12 x \cdot 3 + 18 x + 18 \cdot 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.16.1.3

Add $1$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{3} + 2 x^{2} \cdot 3 - 12 x \cdot x - 12 x \cdot 3 + 18 x + 18 \cdot 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.16.2

Multiply $3$ by $2$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{3} + 6 x^{2} - 12 x \cdot x - 12 x \cdot 3 + 18 x + 18 \cdot 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.16.3

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

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Step 2.2.11.5.19.16.3.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{3} + 6 x^{2} - 12 (x \cdot x) - 12 x \cdot 3 + 18 x + 18 \cdot 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.16.3.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{3} + 6 x^{2} - 12 x^{2} - 12 x \cdot 3 + 18 x + 18 \cdot 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.16.4

Multiply $3$ by $- 12$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{3} + 6 x^{2} - 12 x^{2} - 36 x + 18 x + 18 \cdot 3)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.16.5

Multiply $18$ by $3$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{3} + 6 x^{2} - 12 x^{2} - 36 x + 18 x + 54)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.17

Subtract $12 x^{2}$ from $6 x^{2}$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{3} - 6 x^{2} - 36 x + 18 x + 54)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.19.18

Add $- 36 x$ and $18 x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{3} - 6 x^{2} - 18 x + 54)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.20

Combine the opposite terms in $2 x^{3} + 6 x^{2} - 18 x - 54 + 2 x^{3} - 6 x^{2} - 18 x + 54$ .

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Step 2.2.11.5.20.1

Subtract $6 x^{2}$ from $6 x^{2}$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} - 18 x - 54 + 2 x^{3} + 0 - 18 x + 54)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.20.2

Add $2 x^{3} - 18 x - 54 + 2 x^{3}$ and $0$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} - 18 x - 54 + 2 x^{3} - 18 x + 54)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.20.3

Add $- 54$ and $54$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} - 18 x + 2 x^{3} - 18 x + 0)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.20.4

Add $2 x^{3} - 18 x + 2 x^{3} - 18 x$ and $0$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (2 x^{3} - 18 x + 2 x^{3} - 18 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.21

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (4 x^{3} - 18 x - 18 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.22

Subtract $18 x$ from $- 18 x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x + (- x^{4} + 27 x^{2}) (4 x^{3} - 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.23

Expand $(- x^{4} + 27 x^{2}) (4 x^{3} - 36 x)$ using the FOIL Method.

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Step 2.2.11.5.23.1

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - x^{4} (4 x^{3} - 36 x) + 27 x^{2} (4 x^{3} - 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.23.2

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - x^{4} (4 x^{3}) - x^{4} (- 36 x) + 27 x^{2} (4 x^{3} - 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.23.3

Apply the distributive property.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - x^{4} (4 x^{3}) - x^{4} (- 36 x) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24

Simplify and combine like terms.

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Step 2.2.11.5.24.1

Simplify each term.

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Step 2.2.11.5.24.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 1 \cdot (4 x^{4} x^{3}) - x^{4} (- 36 x) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 2.2.11.5.24.1.2.1

Move $x^{3}$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 1 \cdot (4 (x^{3} x^{4})) - x^{4} (- 36 x) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.2.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 1 \cdot (4 x^{3 + 4}) - x^{4} (- 36 x) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.2.3

Add $3$ and $4$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 1 \cdot (4 x^{7}) - x^{4} (- 36 x) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.3

Multiply $- 1$ by $4$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} - x^{4} (- 36 x) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} - 1 \cdot (- 36 x^{4} x) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.5

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

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Step 2.2.11.5.24.1.5.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} - 1 \cdot (- 36 (x \cdot x^{4})) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.5.2

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

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Step 2.2.11.5.24.1.5.2.1

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} - 1 \cdot (- 36 (x \cdot x^{4})) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.5.2.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} - 1 \cdot (- 36 x^{1 + 4}) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.5.3

Add $1$ and $4$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} - 1 \cdot (- 36 x^{5}) + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.6

Multiply $- 1$ by $- 36$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 27 x^{2} (4 x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.7

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 27 \cdot (4 x^{2} x^{3}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.8

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

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Step 2.2.11.5.24.1.8.1

Move $x^{3}$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 27 \cdot (4 (x^{3} x^{2})) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.8.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 27 \cdot (4 x^{3 + 2}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.8.3

Add $3$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 27 \cdot (4 x^{5}) + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.9

Multiply $27$ by $4$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 108 x^{5} + 27 x^{2} (- 36 x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.10

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 108 x^{5} + 27 \cdot (- 36 x^{2} x)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.11

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

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Step 2.2.11.5.24.1.11.1

Move $x$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 108 x^{5} + 27 \cdot (- 36 (x \cdot x^{2}))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.11.2

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

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Step 2.2.11.5.24.1.11.2.1

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 108 x^{5} + 27 \cdot (- 36 (x \cdot x^{2}))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.11.2.2

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

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 108 x^{5} + 27 \cdot (- 36 x^{1 + 2})}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.11.3

Add $1$ and $2$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 108 x^{5} + 27 \cdot (- 36 x^{3})}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.1.12

Multiply $27$ by $- 36$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 36 x^{5} + 108 x^{5} - 972 x^{3}}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.24.2

Add $36 x^{5}$ and $108 x^{5}$ .

$f'' (x) = \frac{4 x^{7} - 126 x^{5} + 1296 x^{3} - 4374 x - 4 x^{7} + 144 x^{5} - 972 x^{3}}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.25

Subtract $4 x^{7}$ from $4 x^{7}$ .

$f'' (x) = \frac{- 126 x^{5} + 1296 x^{3} - 4374 x + 0 + 144 x^{5} - 972 x^{3}}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.26

Add $- 126 x^{5}$ and $0$ .

$f'' (x) = \frac{- 126 x^{5} + 1296 x^{3} - 4374 x + 144 x^{5} - 972 x^{3}}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.27

Add $- 126 x^{5}$ and $144 x^{5}$ .

$f'' (x) = \frac{18 x^{5} + 1296 x^{3} - 4374 x - 972 x^{3}}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.28

Subtract $972 x^{3}$ from $1296 x^{3}$ .

$f'' (x) = \frac{18 x^{5} + 324 x^{3} - 4374 x}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29

Rewrite $18 x^{5} + 324 x^{3} - 4374 x$ in a factored form.

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Step 2.2.11.5.29.1

Factor $18 x$ out of $18 x^{5} + 324 x^{3} - 4374 x$ .

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Step 2.2.11.5.29.1.1

Factor $18 x$ out of $18 x^{5}$ .

$f'' (x) = \frac{18 x (x^{4}) + 324 x^{3} - 4374 x}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.1.2

Factor $18 x$ out of $324 x^{3}$ .

$f'' (x) = \frac{18 x (x^{4}) + 18 x (18 x^{2}) - 4374 x}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.1.3

Factor $18 x$ out of $- 4374 x$ .

$f'' (x) = \frac{18 x (x^{4}) + 18 x (18 x^{2}) + 18 x (- 243)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.1.4

Factor $18 x$ out of $18 x (x^{4}) + 18 x (18 x^{2})$ .

$f'' (x) = \frac{18 x (x^{4} + 18 x^{2}) + 18 x (- 243)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.1.5

Factor $18 x$ out of $18 x (x^{4} + 18 x^{2}) + 18 x (- 243)$ .

$f'' (x) = \frac{18 x (x^{4} + 18 x^{2} - 243)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f'' (x) = \frac{18 x ({(x^{2})}^{2} + 18 x^{2} - 243)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.3

Let $u_{3} = x^{2}$ . Substitute $u_{3}$ for all occurrences of $x^{2}$ .

$f'' (x) = \frac{18 x ({u_{3}}^{2} + 18 u_{3} - 243)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.4

Factor ${u_{3}}^{2} + 18 u_{3} - 243$ using the AC method.

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Step 2.2.11.5.29.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 243$ and whose sum is $18$ .

$- 9, 27$

Step 2.2.11.5.29.4.2

Write the factored form using these integers.

$f'' (x) = \frac{18 x ((u_{3} - 9) (u_{3} + 27))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.5

Replace all occurrences of $u_{3}$ with $x^{2}$ .

$f'' (x) = \frac{18 x ((x^{2} - 9) (x^{2} + 27))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.6

Rewrite $9$ as $3^{2}$ .

$f'' (x) = \frac{18 x ((x^{2} - 3^{2}) (x^{2} + 27))}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.5.29.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$f'' (x) = \frac{18 x (x + 3) (x - 3) (x^{2} + 27)}{{({(x + 3)}^{2})}^{2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.6

Combine terms.

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Step 2.2.11.6.1

Multiply the exponents in ${({(x + 3)}^{2})}^{2}$ .

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Step 2.2.11.6.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = \frac{18 x (x + 3) (x - 3) (x^{2} + 27)}{{(x + 3)}^{2 \cdot 2} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.6.1.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{18 x (x + 3) (x - 3) (x^{2} + 27)}{{(x + 3)}^{4} {({(x - 3)}^{2})}^{2}}$

Step 2.2.11.6.2

Multiply the exponents in ${({(x - 3)}^{2})}^{2}$ .

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Step 2.2.11.6.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = \frac{18 x (x + 3) (x - 3) (x^{2} + 27)}{{(x + 3)}^{4} {(x - 3)}^{2 \cdot 2}}$

Step 2.2.11.6.2.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{18 x (x + 3) (x - 3) (x^{2} + 27)}{{(x + 3)}^{4} {(x - 3)}^{4}}$

Step 2.2.11.6.3

Cancel the common factor of $x + 3$ and ${(x + 3)}^{4}$ .

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Step 2.2.11.6.3.1

Factor $x + 3$ out of $18 x (x + 3) (x - 3) (x^{2} + 27)$ .

$f'' (x) = \frac{(x + 3) ((18 x (x - 3)) (x^{2} + 27))}{{(x + 3)}^{4} {(x - 3)}^{4}}$

Step 2.2.11.6.3.2

Cancel the common factors.

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Step 2.2.11.6.3.2.1

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

$f'' (x) = \frac{(x + 3) ((18 x (x - 3)) (x^{2} + 27))}{(x + 3) ({(x + 3)}^{3} {(x - 3)}^{4})}$

Step 2.2.11.6.3.2.2

Cancel the common factor.

$f'' (x) = \frac{(x + 3) ((18 x (x - 3)) (x^{2} + 27))}{(x + 3) ({(x + 3)}^{3} {(x - 3)}^{4})}$

Step 2.2.11.6.3.2.3

Rewrite the expression.

$f'' (x) = \frac{(18 x (x - 3)) (x^{2} + 27)}{{(x + 3)}^{3} {(x - 3)}^{4}}$

Step 2.2.11.6.4

Cancel the common factor of $x - 3$ and ${(x - 3)}^{4}$ .

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Step 2.2.11.6.4.1

Factor $x - 3$ out of $18 x (x - 3) (x^{2} + 27)$ .

$f'' (x) = \frac{(x - 3) (18 x (x^{2} + 27))}{{(x + 3)}^{3} {(x - 3)}^{4}}$

Step 2.2.11.6.4.2

Cancel the common factors.

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Step 2.2.11.6.4.2.1

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

$f'' (x) = \frac{(x - 3) (18 x (x^{2} + 27))}{(x - 3) ({(x + 3)}^{3} {(x - 3)}^{3})}$

Step 2.2.11.6.4.2.2

Cancel the common factor.

$f'' (x) = \frac{(x - 3) (18 x (x^{2} + 27))}{(x - 3) ({(x + 3)}^{3} {(x - 3)}^{3})}$

Step 2.2.11.6.4.2.3

Rewrite the expression.

$f'' (x) = \frac{18 x (x^{2} + 27)}{{(x + 3)}^{3} {(x - 3)}^{3}}$

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{18 x (x^{2} + 27)}{{(x + 3)}^{3} {(x - 3)}^{3}}$ .

$\frac{18 x (x^{2} + 27)}{{(x + 3)}^{3} {(x - 3)}^{3}}$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{18 x (x^{2} + 27)}{{(x + 3)}^{3} {(x - 3)}^{3}} = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{18 x (x^{2} + 27)}{{(x + 3)}^{3} {(x - 3)}^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$18 x (x^{2} + 27) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

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

$x = 0$

$x^{2} + 27 = 0$

Step 3.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 3.3.3

Set $x^{2} + 27$ equal to $0$ and solve for $x$ .

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Step 3.3.3.1

Set $x^{2} + 27$ equal to $0$ .

$x^{2} + 27 = 0$

Step 3.3.3.2

Solve $x^{2} + 27 = 0$ for $x$ .

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Step 3.3.3.2.1

Subtract $27$ from both sides of the equation.

$x^{2} = - 27$

Step 3.3.3.2.2

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

$x = \pm \sqrt{- 27}$

Step 3.3.3.2.3

Simplify $\pm \sqrt{- 27}$ .

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Step 3.3.3.2.3.1

Rewrite $- 27$ as $- 1 (27)$ .

$x = \pm \sqrt{- 1 (27)}$

Step 3.3.3.2.3.2

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{27}$

Step 3.3.3.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{27}$

Step 3.3.3.2.3.4

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.3.3.2.3.4.1

Factor $9$ out of $27$ .

$x = \pm i \cdot \sqrt{9 (3)}$

Step 3.3.3.2.3.4.2

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2} \cdot 3}$

Step 3.3.3.2.3.5

Pull terms out from under the radical.

$x = \pm i \cdot (3 \sqrt{3})$

Step 3.3.3.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i \sqrt{3}$

Step 3.3.3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3 i \sqrt{3}$

Step 3.3.3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3 i \sqrt{3}$

Step 3.3.3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3 i \sqrt{3}, - 3 i \sqrt{3}$

Step 3.3.4

The final solution is all the values that make $18 x (x^{2} + 27) = 0$ true.

$x = 0, 3 i \sqrt{3}, - 3 i \sqrt{3}$

Step 4

Find the points where the second derivative is $0$ .

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Step 4.1

Substitute $0$ in $f (x) = \frac{x^{3}}{x^{2} - 9}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \frac{{(0)}^{3}}{{(0)}^{2} - 9}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

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

$f (0) = \frac{0}{0^{2} - 9}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

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

$f (0) = \frac{0}{0 - 9}$

Step 4.1.2.2.2

Subtract $9$ from $0$ .

$f (0) = \frac{0}{- 9}$

Step 4.1.2.3

Divide $0$ by $- 9$ .

$f (0) = 0$

Step 4.1.2.4

The final answer is $0$ .

$0$

Step 4.2

The point found by substituting $0$ in $f (x) = \frac{x^{3}}{x^{2} - 9}$ is $(0, 0)$ . This point can be an inflection point.

$(0, 0)$

Step 5

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, 0) \cup (0, \infty)$

Step 6

Substitute a value from the interval $(- \infty, 0)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.1$ in the expression.

$f'' (- 0.1) = \frac{18 (- 0.1) ({(- 0.1)}^{2} + 27)}{{((- 0.1) + 3)}^{3} {((- 0.1) - 3)}^{3}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $18$ by $- 0.1$ .

$f'' (- 0.1) = \frac{- 1.8 ({(- 0.1)}^{2} + 27)}{{(- 0.1 + 3)}^{3} {(- 0.1 - 3)}^{3}}$

Step 6.2.1.2

Multiply $- 1.8$ by ${(- 0.1)}^{2} + 27$ .

$f'' (- 0.1) = \frac{- 48.618}{{(- 0.1 + 3)}^{3} {(- 0.1 - 3)}^{3}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Add $- 0.1$ and $3$ .

$f'' (- 0.1) = \frac{- 48.618}{{2.9}^{3} {(- 0.1 - 3)}^{3}}$

Step 6.2.2.2

Subtract $3$ from $- 0.1$ .

$f'' (- 0.1) = \frac{- 48.618}{{2.9}^{3} {(- 3.1)}^{3}}$

Step 6.2.2.3

Raise $2.9$ to the power of $3$ .

$f'' (- 0.1) = \frac{- 48.618}{24.389 {(- 3.1)}^{3}}$

Step 6.2.2.4

Raise $- 3.1$ to the power of $3$ .

$f'' (- 0.1) = \frac{- 48.618}{24.389 \cdot - 29.791}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Multiply $24.389$ by $- 29.791$ .

$f'' (- 0.1) = \frac{- 48.618}{- 726.572699}$

Step 6.2.3.2

Divide $- 48.618$ by $- 726.572699$ .

$f'' (- 0.1) = 0.06691415$

Step 6.2.4

The final answer is $0.06691415$ .

$0.06691415$

Step 6.3

At $- 0.1$ , the second derivative is $0.06691415$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 0)$ .

Increasing on $(- \infty, 0)$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(0, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $0.1$ in the expression.

$f'' (0.1) = \frac{18 (0.1) ({(0.1)}^{2} + 27)}{{((0.1) + 3)}^{3} {((0.1) - 3)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Multiply $18$ by $0.1$ .

$f'' (0.1) = \frac{1.8 ({0.1}^{2} + 27)}{{(0.1 + 3)}^{3} {(0.1 - 3)}^{3}}$

Step 7.2.1.2

Multiply $1.8$ by ${0.1}^{2} + 27$ .

$f'' (0.1) = \frac{48.618}{{(0.1 + 3)}^{3} {(0.1 - 3)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Add $0.1$ and $3$ .

$f'' (0.1) = \frac{48.618}{{3.1}^{3} {(0.1 - 3)}^{3}}$

Step 7.2.2.2

Subtract $3$ from $0.1$ .

$f'' (0.1) = \frac{48.618}{{3.1}^{3} {(- 2.9)}^{3}}$

Step 7.2.2.3

Raise $3.1$ to the power of $3$ .

$f'' (0.1) = \frac{48.618}{29.791 {(- 2.9)}^{3}}$

Step 7.2.2.4

Raise $- 2.9$ to the power of $3$ .

$f'' (0.1) = \frac{48.618}{29.791 \cdot - 24.389}$

Step 7.2.3

Simplify the expression.

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Step 7.2.3.1

Multiply $29.791$ by $- 24.389$ .

$f'' (0.1) = \frac{48.618}{- 726.572699}$

Step 7.2.3.2

Divide $48.618$ by $- 726.572699$ .

$f'' (0.1) = - 0.06691415$

Step 7.2.4

The final answer is $- 0.06691415$ .

$- 0.06691415$

Step 7.3

At $0.1$ , the second derivative is $- 0.06691415$ . Since this is negative, the second derivative is decreasing on the interval $(0, \infty)$

Decreasing on $(0, \infty)$ since $f'' (x) < 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(0, 0)$ .

$(0, 0)$

Step 9