Calculus Examples

$y = \frac{x^{4}}{x^{4} - 81}$

Step 1

Write $y = \frac{x^{4}}{x^{4} - 81}$ as a function.

$f (x) = \frac{x^{4}}{x^{4} - 81}$

Step 2

Find the first derivative of the function.

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Step 2.1

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 $f (x) = x^{4}$ and $g (x) = x^{4} - 81$ .

$\frac{(x^{4} - 81) \frac{d}{d x} [x^{4}] - x^{4} \frac{d}{d x} [x^{4} - 81]}{{(x^{4} - 81)}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

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

$\frac{(x^{4} - 81) (4 x^{3}) - x^{4} \frac{d}{d x} [x^{4} - 81]}{{(x^{4} - 81)}^{2}}$

Step 2.2.2

Move $4$ to the left of $x^{4} - 81$ .

$\frac{4 \cdot (x^{4} - 81) x^{3} - x^{4} \frac{d}{d x} [x^{4} - 81]}{{(x^{4} - 81)}^{2}}$

Step 2.2.3

By the Sum Rule, the derivative of $x^{4} - 81$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 81]$ .

$\frac{4 (x^{4} - 81) x^{3} - x^{4} (\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 81])}{{(x^{4} - 81)}^{2}}$

Step 2.2.4

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

$\frac{4 (x^{4} - 81) x^{3} - x^{4} (4 x^{3} + \frac{d}{d x} [- 81])}{{(x^{4} - 81)}^{2}}$

Step 2.2.5

Since $- 81$ is constant with respect to $x$ , the derivative of $- 81$ with respect to $x$ is $0$ .

$\frac{4 (x^{4} - 81) x^{3} - x^{4} (4 x^{3} + 0)}{{(x^{4} - 81)}^{2}}$

Step 2.2.6

Simplify the expression.

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Step 2.2.6.1

Add $4 x^{3}$ and $0$ .

$\frac{4 (x^{4} - 81) x^{3} - x^{4} (4 x^{3})}{{(x^{4} - 81)}^{2}}$

Step 2.2.6.2

Multiply $4$ by $- 1$ .

$\frac{4 (x^{4} - 81) x^{3} - 4 x^{4} x^{3}}{{(x^{4} - 81)}^{2}}$

Step 2.3

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

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Step 2.3.1

Move $x^{3}$ .

$\frac{4 (x^{4} - 81) x^{3} - 4 (x^{3} x^{4})}{{(x^{4} - 81)}^{2}}$

Step 2.3.2

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

$\frac{4 (x^{4} - 81) x^{3} - 4 x^{3 + 4}}{{(x^{4} - 81)}^{2}}$

Step 2.3.3

Add $3$ and $4$ .

$\frac{4 (x^{4} - 81) x^{3} - 4 x^{7}}{{(x^{4} - 81)}^{2}}$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$\frac{(4 x^{4} + 4 \cdot - 81) x^{3} - 4 x^{7}}{{(x^{4} - 81)}^{2}}$

Step 2.4.2

Apply the distributive property.

$\frac{4 x^{4} x^{3} + 4 \cdot - 81 x^{3} - 4 x^{7}}{{(x^{4} - 81)}^{2}}$

Step 2.4.3

Simplify the numerator.

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

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

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Step 2.4.3.1.1.1

Move $x^{3}$ .

$\frac{4 (x^{3} x^{4}) + 4 \cdot - 81 x^{3} - 4 x^{7}}{{(x^{4} - 81)}^{2}}$

Step 2.4.3.1.1.2

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

$\frac{4 x^{3 + 4} + 4 \cdot - 81 x^{3} - 4 x^{7}}{{(x^{4} - 81)}^{2}}$

Step 2.4.3.1.1.3

Add $3$ and $4$ .

$\frac{4 x^{7} + 4 \cdot - 81 x^{3} - 4 x^{7}}{{(x^{4} - 81)}^{2}}$

Step 2.4.3.1.2

Multiply $4$ by $- 81$ .

$\frac{4 x^{7} - 324 x^{3} - 4 x^{7}}{{(x^{4} - 81)}^{2}}$

Step 2.4.3.2

Combine the opposite terms in $4 x^{7} - 324 x^{3} - 4 x^{7}$ .

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Step 2.4.3.2.1

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

$\frac{- 324 x^{3} + 0}{{(x^{4} - 81)}^{2}}$

Step 2.4.3.2.2

Add $- 324 x^{3}$ and $0$ .

$\frac{- 324 x^{3}}{{(x^{4} - 81)}^{2}}$

Step 2.4.4

Move the negative in front of the fraction.

$- \frac{324 x^{3}}{{(x^{4} - 81)}^{2}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Since $- 324$ is constant with respect to $x$ , the derivative of $- \frac{324 x^{3}}{{(x^{4} - 81)}^{2}}$ with respect to $x$ is $- 324 \frac{d}{d x} [\frac{x^{3}}{{(x^{4} - 81)}^{2}}]$ .

$f'' (x) = - 324 \frac{d}{d x} \frac{x^{3}}{{(x^{4} - 81)}^{2}}$

Step 3.2

$f'' (x) = - 324 \frac{{(x^{4} - 81)}^{2} \frac{d}{d x} (x^{3}) - x^{3} \frac{d}{d x} {(x^{4} - 81)}^{2}}{{({(x^{4} - 81)}^{2})}^{2}}$

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

Multiply the exponents in ${({(x^{4} - 81)}^{2})}^{2}$ .

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Step 3.3.1.1

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

$f'' (x) = - 324 \frac{{(x^{4} - 81)}^{2} \frac{d}{d x} (x^{3}) - x^{3} \frac{d}{d x} {(x^{4} - 81)}^{2}}{{(x^{4} - 81)}^{2 \cdot 2}}$

Step 3.3.1.2

Multiply $2$ by $2$ .

$f'' (x) = - 324 \frac{{(x^{4} - 81)}^{2} \frac{d}{d x} (x^{3}) - x^{3} \frac{d}{d x} {(x^{4} - 81)}^{2}}{{(x^{4} - 81)}^{4}}$

Step 3.3.2

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

$f'' (x) = - 324 \frac{{(x^{4} - 81)}^{2} (3 x^{2}) - x^{3} \frac{d}{d x} {(x^{4} - 81)}^{2}}{{(x^{4} - 81)}^{4}}$

Step 3.3.3

Move $3$ to the left of ${(x^{4} - 81)}^{2}$ .

$f'' (x) = - 324 \frac{3 \cdot ({(x^{4} - 81)}^{2} x^{2}) - x^{3} \frac{d}{d x} {(x^{4} - 81)}^{2}}{{(x^{4} - 81)}^{4}}$

Step 3.4

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 $g (x) = x^{4} - 81$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{4} - 81$ .

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - x^{3} (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{4} - 81))}{{(x^{4} - 81)}^{4}}$

Step 3.4.2

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

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - x^{3} (2 u \frac{d}{d x} (x^{4} - 81))}{{(x^{4} - 81)}^{4}}$

Step 3.4.3

Replace all occurrences of $u$ with $x^{4} - 81$ .

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - x^{3} (2 (x^{4} - 81) \frac{d}{d x} (x^{4} - 81))}{{(x^{4} - 81)}^{4}}$

Step 3.5

Differentiate.

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Step 3.5.1

Multiply $2$ by $- 1$ .

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - 2 x^{3} ((x^{4} - 81) \frac{d}{d x} (x^{4} - 81))}{{(x^{4} - 81)}^{4}}$

Step 3.5.2

By the Sum Rule, the derivative of $x^{4} - 81$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 81]$ .

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - 2 x^{3} ((x^{4} - 81) (\frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 81)))}{{(x^{4} - 81)}^{4}}$

Step 3.5.3

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

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - 2 x^{3} ((x^{4} - 81) (4 x^{3} + \frac{d}{d x} (- 81)))}{{(x^{4} - 81)}^{4}}$

Step 3.5.4

Since $- 81$ is constant with respect to $x$ , the derivative of $- 81$ with respect to $x$ is $0$ .

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - 2 x^{3} ((x^{4} - 81) (4 x^{3} + 0))}{{(x^{4} - 81)}^{4}}$

Step 3.5.5

Simplify the expression.

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Step 3.5.5.1

Add $4 x^{3}$ and $0$ .

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - 2 x^{3} ((x^{4} - 81) (4 x^{3}))}{{(x^{4} - 81)}^{4}}$

Step 3.5.5.2

Move $4$ to the left of $x^{4} - 81$ .

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - 2 x^{3} (4 \cdot ((x^{4} - 81) x^{3}))}{{(x^{4} - 81)}^{4}}$

Step 3.5.5.3

Multiply $4$ by $- 2$ .

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - 8 x^{3} ((x^{4} - 81) x^{3})}{{(x^{4} - 81)}^{4}}$

Step 3.6

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

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - 8 x^{3 + 3} (x^{4} - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.7

Add $3$ and $3$ .

$f'' (x) = - 324 \frac{3 {(x^{4} - 81)}^{2} x^{2} - 8 x^{6} (x^{4} - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.8

Combine $- 324$ and $\frac{3 {(x^{4} - 81)}^{2} x^{2} - 8 x^{6} (x^{4} - 81)}{{(x^{4} - 81)}^{4}}$ .

$f'' (x) = \frac{- 324 (3 {(x^{4} - 81)}^{2} x^{2} - 8 x^{6} (x^{4} - 81))}{{(x^{4} - 81)}^{4}}$

Step 3.9

Move the negative in front of the fraction.

$f'' (x) = - \frac{(324) (3 {(x^{4} - 81)}^{2} x^{2} - 8 x^{6} (x^{4} - 81))}{{(x^{4} - 81)}^{4}}$

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

$f'' (x) = - \frac{324 (3 {(x^{4} - 81)}^{2} x^{2} - 8 x^{6} x^{4} - 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.2

Apply the distributive property.

$f'' (x) = - \frac{324 (3 {(x^{4} - 81)}^{2} x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3

Simplify the numerator.

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Step 3.10.3.1

Simplify each term.

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Step 3.10.3.1.1

Rewrite ${(x^{4} - 81)}^{2}$ as $(x^{4} - 81) (x^{4} - 81)$ .

$f'' (x) = - \frac{324 (3 ((x^{4} - 81) (x^{4} - 81)) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.2

Expand $(x^{4} - 81) (x^{4} - 81)$ using the FOIL Method.

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Step 3.10.3.1.2.1

Apply the distributive property.

$f'' (x) = - \frac{324 (3 (x^{4} (x^{4} - 81) - 81 (x^{4} - 81)) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.2.2

Apply the distributive property.

$f'' (x) = - \frac{324 (3 (x^{4} x^{4} + x^{4} \cdot - 81 - 81 (x^{4} - 81)) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.2.3

Apply the distributive property.

$f'' (x) = - \frac{324 (3 (x^{4} x^{4} + x^{4} \cdot - 81 - 81 x^{4} - 81 \cdot - 81) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.3

Simplify and combine like terms.

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Step 3.10.3.1.3.1

Simplify each term.

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Step 3.10.3.1.3.1.1

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

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Step 3.10.3.1.3.1.1.1

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

$f'' (x) = - \frac{324 (3 (x^{4 + 4} + x^{4} \cdot - 81 - 81 x^{4} - 81 \cdot - 81) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.3.1.1.2

Add $4$ and $4$ .

$f'' (x) = - \frac{324 (3 (x^{8} + x^{4} \cdot - 81 - 81 x^{4} - 81 \cdot - 81) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.3.1.2

Move $- 81$ to the left of $x^{4}$ .

$f'' (x) = - \frac{324 (3 (x^{8} - 81 \cdot x^{4} - 81 x^{4} - 81 \cdot - 81) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.3.1.3

Multiply $- 81$ by $- 81$ .

$f'' (x) = - \frac{324 (3 (x^{8} - 81 x^{4} - 81 x^{4} + 6561) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.3.2

Subtract $81 x^{4}$ from $- 81 x^{4}$ .

$f'' (x) = - \frac{324 (3 (x^{8} - 162 x^{4} + 6561) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.4

Apply the distributive property.

$f'' (x) = - \frac{324 ((3 x^{8} + 3 (- 162 x^{4}) + 3 \cdot 6561) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.5

Simplify.

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Step 3.10.3.1.5.1

Multiply $- 162$ by $3$ .

$f'' (x) = - \frac{324 ((3 x^{8} - 486 x^{4} + 3 \cdot 6561) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.5.2

Multiply $3$ by $6561$ .

$f'' (x) = - \frac{324 ((3 x^{8} - 486 x^{4} + 19683) x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.6

Apply the distributive property.

$f'' (x) = - \frac{324 (3 x^{8} x^{2} - 486 x^{4} x^{2} + 19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.7

Simplify.

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Step 3.10.3.1.7.1

Multiply $x^{8}$ by $x^{2}$ by adding the exponents.

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Step 3.10.3.1.7.1.1

Move $x^{2}$ .

$f'' (x) = - \frac{324 (3 (x^{2} x^{8}) - 486 x^{4} x^{2} + 19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.7.1.2

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

$f'' (x) = - \frac{324 (3 x^{2 + 8} - 486 x^{4} x^{2} + 19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.7.1.3

Add $2$ and $8$ .

$f'' (x) = - \frac{324 (3 x^{10} - 486 x^{4} x^{2} + 19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.7.2

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 3.10.3.1.7.2.1

Move $x^{2}$ .

$f'' (x) = - \frac{324 (3 x^{10} - 486 (x^{2} x^{4}) + 19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.7.2.2

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

$f'' (x) = - \frac{324 (3 x^{10} - 486 x^{2 + 4} + 19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.7.2.3

Add $2$ and $4$ .

$f'' (x) = - \frac{324 (3 x^{10} - 486 x^{6} + 19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.8

Apply the distributive property.

$f'' (x) = - \frac{324 (3 x^{10}) + 324 (- 486 x^{6}) + 324 (19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.9

Simplify.

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Step 3.10.3.1.9.1

Multiply $3$ by $324$ .

$f'' (x) = - \frac{972 x^{10} + 324 (- 486 x^{6}) + 324 (19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.9.2

Multiply $- 486$ by $324$ .

$f'' (x) = - \frac{972 x^{10} - 157464 x^{6} + 324 (19683 x^{2}) + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.9.3

Multiply $19683$ by $324$ .

$f'' (x) = - \frac{972 x^{10} - 157464 x^{6} + 6377292 x^{2} + 324 (- 8 x^{6} x^{4}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.10

Multiply $x^{6}$ by $x^{4}$ by adding the exponents.

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Step 3.10.3.1.10.1

Move $x^{4}$ .

$f'' (x) = - \frac{972 x^{10} - 157464 x^{6} + 6377292 x^{2} + 324 (- 8 (x^{4} x^{6})) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.10.2

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

$f'' (x) = - \frac{972 x^{10} - 157464 x^{6} + 6377292 x^{2} + 324 (- 8 x^{4 + 6}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.10.3

Add $4$ and $6$ .

$f'' (x) = - \frac{972 x^{10} - 157464 x^{6} + 6377292 x^{2} + 324 (- 8 x^{10}) + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.11

Multiply $- 8$ by $324$ .

$f'' (x) = - \frac{972 x^{10} - 157464 x^{6} + 6377292 x^{2} - 2592 x^{10} + 324 (- 8 x^{6} \cdot - 81)}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.12

Multiply $- 81$ by $- 8$ .

$f'' (x) = - \frac{972 x^{10} - 157464 x^{6} + 6377292 x^{2} - 2592 x^{10} + 324 (648 x^{6})}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.1.13

Multiply $648$ by $324$ .

$f'' (x) = - \frac{972 x^{10} - 157464 x^{6} + 6377292 x^{2} - 2592 x^{10} + 209952 x^{6}}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.2

Subtract $2592 x^{10}$ from $972 x^{10}$ .

$f'' (x) = - \frac{- 1620 x^{10} - 157464 x^{6} + 6377292 x^{2} + 209952 x^{6}}{{(x^{4} - 81)}^{4}}$

Step 3.10.3.3

Add $- 157464 x^{6}$ and $209952 x^{6}$ .

$f'' (x) = - \frac{- 1620 x^{10} + 52488 x^{6} + 6377292 x^{2}}{{(x^{4} - 81)}^{4}}$

Step 3.10.4

Simplify the numerator.

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Step 3.10.4.1

Factor $324 x^{2}$ out of $- 1620 x^{10} + 52488 x^{6} + 6377292 x^{2}$ .

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Step 3.10.4.1.1

Factor $324 x^{2}$ out of $- 1620 x^{10}$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{8}) + 52488 x^{6} + 6377292 x^{2}}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.1.2

Factor $324 x^{2}$ out of $52488 x^{6}$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{8}) + 324 x^{2} (162 x^{4}) + 6377292 x^{2}}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.1.3

Factor $324 x^{2}$ out of $6377292 x^{2}$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{8}) + 324 x^{2} (162 x^{4}) + 324 x^{2} (19683)}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.1.4

Factor $324 x^{2}$ out of $324 x^{2} (- 5 x^{8}) + 324 x^{2} (162 x^{4})$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{8} + 162 x^{4}) + 324 x^{2} (19683)}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.1.5

Factor $324 x^{2}$ out of $324 x^{2} (- 5 x^{8} + 162 x^{4}) + 324 x^{2} (19683)$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{8} + 162 x^{4} + 19683)}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.2

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$f'' (x) = - \frac{324 x^{2} (- 5 {(x^{4})}^{2} + 162 x^{4} + 19683)}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.3

Let $u = x^{4}$ . Substitute $u$ for all occurrences of $x^{4}$ .

$f'' (x) = - \frac{324 x^{2} (- 5 u^{2} + 162 u + 19683)}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.4

Factor by grouping.

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Step 3.10.4.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 5 \cdot 19683 = - 98415$ and whose sum is $b = 162$ .

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Step 3.10.4.4.1.1

Factor $162$ out of $162 u$ .

$f'' (x) = - \frac{324 x^{2} (- 5 u^{2} + 162 (u) + 19683)}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.4.1.2

Rewrite $162$ as $- 243$ plus $405$

$f'' (x) = - \frac{324 x^{2} (- 5 u^{2} + (- 243 + 405) u + 19683)}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.4.1.3

Apply the distributive property.

$f'' (x) = - \frac{324 x^{2} (- 5 u^{2} - 243 u + 405 u + 19683)}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.4.2

Factor out the greatest common factor from each group.

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Step 3.10.4.4.2.1

Group the first two terms and the last two terms.

$f'' (x) = - \frac{324 x^{2} ((- 5 u^{2} - 243 u) + 405 u + 19683)}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.4.2.2

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

$f'' (x) = - \frac{324 x^{2} (u (- 5 u - 243) - 81 (- 5 u - 243))}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.4.3

Factor the polynomial by factoring out the greatest common factor, $- 5 u - 243$ .

$f'' (x) = - \frac{324 x^{2} ((- 5 u - 243) (u - 81))}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.5

Replace all occurrences of $u$ with $x^{4}$ .

$f'' (x) = - \frac{324 x^{2} ((- 5 x^{4} - 243) (x^{4} - 81))}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.6

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

$f'' (x) = - \frac{324 x^{2} ((- 5 x^{4} - 243) ({(x^{2})}^{2} - 81))}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.7

Rewrite $81$ as $9^{2}$ .

$f'' (x) = - \frac{324 x^{2} ((- 5 x^{4} - 243) ({(x^{2})}^{2} - 9^{2}))}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 9$ .

$f'' (x) = - \frac{324 x^{2} ((- 5 x^{4} - 243) ((x^{2} + 9) (x^{2} - 9)))}{{(x^{4} - 81)}^{4}}$

Step 3.10.4.9

Factor.

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{4} - 81)}^{4}}$

Step 3.10.5

Simplify the denominator.

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Step 3.10.5.1

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

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{({(x^{2})}^{2} - 81)}^{4}}$

Step 3.10.5.2

Rewrite $81$ as $9^{2}$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{({(x^{2})}^{2} - 9^{2})}^{4}}$

Step 3.10.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 9$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{((x^{2} + 9) (x^{2} - 9))}^{4}}$

Step 3.10.5.4

Simplify.

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Step 3.10.5.4.1

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

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{((x^{2} + 9) (x^{2} - 3^{2}))}^{4}}$

Step 3.10.5.4.2

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 $b = 3$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{((x^{2} + 9) (x + 3) (x - 3))}^{4}}$

Step 3.10.5.5

Apply the product rule to $(x^{2} + 9) (x + 3) (x - 3)$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{((x^{2} + 9) (x + 3))}^{4} {(x - 3)}^{4}}$

Step 3.10.5.6

Expand $(x^{2} + 9) (x + 3)$ using the FOIL Method.

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Step 3.10.5.6.1

Apply the distributive property.

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{2} (x + 3) + 9 (x + 3))}^{4} {(x - 3)}^{4}}$

Step 3.10.5.6.2

Apply the distributive property.

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{2} x + x^{2} \cdot 3 + 9 (x + 3))}^{4} {(x - 3)}^{4}}$

Step 3.10.5.6.3

Apply the distributive property.

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{2} x + x^{2} \cdot 3 + 9 x + 9 \cdot 3)}^{4} {(x - 3)}^{4}}$

Step 3.10.5.7

Simplify each term.

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Step 3.10.5.7.1

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

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Step 3.10.5.7.1.1

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

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Step 3.10.5.7.1.1.1

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

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{2} x + x^{2} \cdot 3 + 9 x + 9 \cdot 3)}^{4} {(x - 3)}^{4}}$

Step 3.10.5.7.1.1.2

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

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{2 + 1} + x^{2} \cdot 3 + 9 x + 9 \cdot 3)}^{4} {(x - 3)}^{4}}$

Step 3.10.5.7.1.2

Add $2$ and $1$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{3} + x^{2} \cdot 3 + 9 x + 9 \cdot 3)}^{4} {(x - 3)}^{4}}$

Step 3.10.5.7.2

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

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{3} + 3 \cdot x^{2} + 9 x + 9 \cdot 3)}^{4} {(x - 3)}^{4}}$

Step 3.10.5.7.3

Multiply $9$ by $3$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{3} + 3 x^{2} + 9 x + 27)}^{4} {(x - 3)}^{4}}$

Step 3.10.5.8

Factor out the greatest common factor from each group.

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Step 3.10.5.8.1

Group the first two terms and the last two terms.

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{((x^{3} + 3 x^{2}) + 9 x + 27)}^{4} {(x - 3)}^{4}}$

Step 3.10.5.8.2

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

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x^{2} (x + 3) + 9 (x + 3))}^{4} {(x - 3)}^{4}}$

Step 3.10.5.9

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

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{((x + 3) (x^{2} + 9))}^{4} {(x - 3)}^{4}}$

Step 3.10.5.10

Apply the product rule to $(x + 3) (x^{2} + 9)$ .

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)}{{(x + 3)}^{4} {(x^{2} + 9)}^{4} {(x - 3)}^{4}}$

Step 3.10.6

Cancel the common factor of $x^{2} + 9$ and ${(x^{2} + 9)}^{4}$ .

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Step 3.10.6.1

Factor $x^{2} + 9$ out of $324 x^{2} (- 5 x^{4} - 243) (x^{2} + 9) (x + 3) (x - 3)$ .

$f'' (x) = - \frac{(x^{2} + 9) ((324 x^{2} (- 5 x^{4} - 243) (x + 3)) (x - 3))}{{(x + 3)}^{4} {(x^{2} + 9)}^{4} {(x - 3)}^{4}}$

Step 3.10.6.2

Cancel the common factors.

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Step 3.10.6.2.1

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

$f'' (x) = - \frac{(x^{2} + 9) ((324 x^{2} (- 5 x^{4} - 243) (x + 3)) (x - 3))}{(x^{2} + 9) ({(x + 3)}^{4} {(x^{2} + 9)}^{3} {(x - 3)}^{4})}$

Step 3.10.6.2.2

Cancel the common factor.

$f'' (x) = - \frac{(x^{2} + 9) ((324 x^{2} (- 5 x^{4} - 243) (x + 3)) (x - 3))}{(x^{2} + 9) ({(x + 3)}^{4} {(x^{2} + 9)}^{3} {(x - 3)}^{4})}$

Step 3.10.6.2.3

Rewrite the expression.

$f'' (x) = - \frac{(324 x^{2} (- 5 x^{4} - 243) (x + 3)) (x - 3)}{{(x + 3)}^{4} {(x^{2} + 9)}^{3} {(x - 3)}^{4}}$

Step 3.10.7

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

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Step 3.10.7.1

Factor $x + 3$ out of $(324 x^{2} (- 5 x^{4} - 243) (x + 3)) (x - 3)$ .

$f'' (x) = - \frac{(x + 3) ((324 x^{2} (- 5 x^{4} - 243)) (x - 3))}{{(x + 3)}^{4} {(x^{2} + 9)}^{3} {(x - 3)}^{4}}$

Step 3.10.7.2

Cancel the common factors.

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Step 3.10.7.2.1

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

$f'' (x) = - \frac{(x + 3) ((324 x^{2} (- 5 x^{4} - 243)) (x - 3))}{(x + 3) ({(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{4})}$

Step 3.10.7.2.2

Cancel the common factor.

$f'' (x) = - \frac{(x + 3) ((324 x^{2} (- 5 x^{4} - 243)) (x - 3))}{(x + 3) ({(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{4})}$

Step 3.10.7.2.3

Rewrite the expression.

$f'' (x) = - \frac{(324 x^{2} (- 5 x^{4} - 243)) (x - 3)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{4}}$

Step 3.10.8

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

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Step 3.10.8.1

Factor $x - 3$ out of $(324 x^{2} (- 5 x^{4} - 243)) (x - 3)$ .

$f'' (x) = - \frac{(x - 3) (324 x^{2} (- 5 x^{4} - 243))}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{4}}$

Step 3.10.8.2

Cancel the common factors.

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Step 3.10.8.2.1

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

$f'' (x) = - \frac{(x - 3) (324 x^{2} (- 5 x^{4} - 243))}{(x - 3) ({(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3})}$

Step 3.10.8.2.2

Cancel the common factor.

$f'' (x) = - \frac{(x - 3) (324 x^{2} (- 5 x^{4} - 243))}{(x - 3) ({(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3})}$

Step 3.10.8.2.3

Rewrite the expression.

$f'' (x) = - \frac{324 x^{2} (- 5 x^{4} - 243)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$

Step 3.10.9

Factor $- 1$ out of $- 5 x^{4}$ .

$f'' (x) = - \frac{324 x^{2} (- (5 x^{4}) - 243)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$

Step 3.10.10

Rewrite $- 243$ as $- 1 (243)$ .

$f'' (x) = - \frac{324 x^{2} (- (5 x^{4}) - 1 \cdot 243)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$

Step 3.10.11

Factor $- 1$ out of $- (5 x^{4}) - 1 (243)$ .

$f'' (x) = - \frac{324 x^{2} (- (5 x^{4} + 243))}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$

Step 3.10.12

Rewrite $- (5 x^{4} + 243)$ as $- 1 (5 x^{4} + 243)$ .

$f'' (x) = - \frac{324 x^{2} (- 1 (5 x^{4} + 243))}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$

Step 3.10.13

Move the negative in front of the fraction.

$f'' (x) = \frac{(324 x^{2}) (5 x^{4} + 243)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$

Step 3.10.14

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{(324 x^{2}) (5 x^{4} + 243)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}})$

Step 3.10.15

Multiply $\frac{(324 x^{2}) (5 x^{4} + 243)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$ by $1$ .

$f'' (x) = \frac{(324 x^{2}) (5 x^{4} + 243)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$

Step 3.10.16

Reorder factors in $\frac{(324 x^{2}) (5 x^{4} + 243)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$ .

$f'' (x) = \frac{324 x^{2} (5 x^{4} + 243)}{{(x + 3)}^{3} {(x^{2} + 9)}^{3} {(x - 3)}^{3}}$

Step 4

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

$- \frac{324 x^{3}}{{(x^{4} - 81)}^{2}} = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

$f' (x) = \frac{(x^{4} - 81) \frac{d}{d x} (x^{4}) - x^{4} \frac{d}{d x} (x^{4} - 81)}{{(x^{4} - 81)}^{2}}$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

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

$f' (x) = \frac{(x^{4} - 81) (4 x^{3}) - x^{4} \frac{d}{d x} (x^{4} - 81)}{{(x^{4} - 81)}^{2}}$

Step 5.1.2.2

Move $4$ to the left of $x^{4} - 81$ .

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

Step 5.1.2.3

By the Sum Rule, the derivative of $x^{4} - 81$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 81]$ .

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

Step 5.1.2.4

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

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

Step 5.1.2.5

Since $- 81$ is constant with respect to $x$ , the derivative of $- 81$ with respect to $x$ is $0$ .

$f' (x) = \frac{4 (x^{4} - 81) x^{3} - x^{4} (4 x^{3} + 0)}{{(x^{4} - 81)}^{2}}$

Step 5.1.2.6

Simplify the expression.

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Step 5.1.2.6.1

Add $4 x^{3}$ and $0$ .

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

Step 5.1.2.6.2

Multiply $4$ by $- 1$ .

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

Step 5.1.3

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

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Step 5.1.3.1

Move $x^{3}$ .

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

Step 5.1.3.2

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

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

Step 5.1.3.3

Add $3$ and $4$ .

$f' (x) = \frac{4 (x^{4} - 81) x^{3} - 4 x^{7}}{{(x^{4} - 81)}^{2}}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Apply the distributive property.

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

Step 5.1.4.2

Apply the distributive property.

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

Step 5.1.4.3

Simplify the numerator.

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Step 5.1.4.3.1

Simplify each term.

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Step 5.1.4.3.1.1

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

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Step 5.1.4.3.1.1.1

Move $x^{3}$ .

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

Step 5.1.4.3.1.1.2

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

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

Step 5.1.4.3.1.1.3

Add $3$ and $4$ .

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

Step 5.1.4.3.1.2

Multiply $4$ by $- 81$ .

$f' (x) = \frac{4 x^{7} - 324 x^{3} - 4 x^{7}}{{(x^{4} - 81)}^{2}}$

Step 5.1.4.3.2

Combine the opposite terms in $4 x^{7} - 324 x^{3} - 4 x^{7}$ .

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Step 5.1.4.3.2.1

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

$f' (x) = \frac{- 324 x^{3} + 0}{{(x^{4} - 81)}^{2}}$

Step 5.1.4.3.2.2

Add $- 324 x^{3}$ and $0$ .

$f' (x) = \frac{- 324 x^{3}}{{(x^{4} - 81)}^{2}}$

Step 5.1.4.4

Move the negative in front of the fraction.

$f' (x) = - \frac{324 x^{3}}{{(x^{4} - 81)}^{2}}$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{324 x^{3}}{{(x^{4} - 81)}^{2}}$ .

$- \frac{324 x^{3}}{{(x^{4} - 81)}^{2}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{324 x^{3}}{{(x^{4} - 81)}^{2}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{324 x^{3}}{{(x^{4} - 81)}^{2}} = 0$

Step 6.2

Set the numerator equal to zero.

$324 x^{3} = 0$

Step 6.3

Solve the equation for $x$ .

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Step 6.3.1

Divide each term in $324 x^{3} = 0$ by $324$ and simplify.

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Step 6.3.1.1

Divide each term in $324 x^{3} = 0$ by $324$ .

$\frac{324 x^{3}}{324} = \frac{0}{324}$

Step 6.3.1.2

Simplify the left side.

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Step 6.3.1.2.1

Cancel the common factor of $324$ .

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Step 6.3.1.2.1.1

Cancel the common factor.

$\frac{324 x^{3}}{324} = \frac{0}{324}$

Step 6.3.1.2.1.2

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

$x^{3} = \frac{0}{324}$

Step 6.3.1.3

Simplify the right side.

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Step 6.3.1.3.1

Divide $0$ by $324$ .

$x^{3} = 0$

Step 6.3.2

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

$x = \sqrt[3]{0}$

Step 6.3.3

Simplify $\sqrt[3]{0}$ .

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Step 6.3.3.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 6.3.3.2

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

$x = 0$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

Set the denominator in $\frac{324 x^{3}}{{(x^{4} - 81)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x^{4} - 81)}^{2} = 0$

Step 7.2

Solve for $x$ .

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Step 7.2.1

Factor the left side of the equation.

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Step 7.2.1.1

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

${({(x^{2})}^{2} - 81)}^{2} = 0$

Step 7.2.1.2

Rewrite $81$ as $9^{2}$ .

${({(x^{2})}^{2} - 9^{2})}^{2} = 0$

Step 7.2.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 9$ .

${((x^{2} + 9) (x^{2} - 9))}^{2} = 0$

Step 7.2.1.4

Simplify.

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Step 7.2.1.4.1

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

${((x^{2} + 9) (x^{2} - 3^{2}))}^{2} = 0$

Step 7.2.1.4.2

Factor.

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Step 7.2.1.4.2.1

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 $b = 3$ .

${((x^{2} + 9) ((x + 3) (x - 3)))}^{2} = 0$

Step 7.2.1.4.2.2

Remove unnecessary parentheses.

${((x^{2} + 9) (x + 3) (x - 3))}^{2} = 0$

Step 7.2.1.5

Apply the product rule to $(x^{2} + 9) (x + 3) (x - 3)$ .

${((x^{2} + 9) (x + 3))}^{2} {(x - 3)}^{2} = 0$

Step 7.2.1.6

Apply the product rule to $(x^{2} + 9) (x + 3)$ .

${(x^{2} + 9)}^{2} {(x + 3)}^{2} {(x - 3)}^{2} = 0$

Step 7.2.2

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

${(x^{2} + 9)}^{2} = 0$

${(x + 3)}^{2} = 0$

${(x - 3)}^{2} = 0$

Step 7.2.3

Set ${(x^{2} + 9)}^{2}$ equal to $0$ and solve for $x$ .

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Step 7.2.3.1

Set ${(x^{2} + 9)}^{2}$ equal to $0$ .

${(x^{2} + 9)}^{2} = 0$

Step 7.2.3.2

Solve ${(x^{2} + 9)}^{2} = 0$ for $x$ .

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Step 7.2.3.2.1

Set the $x^{2} + 9$ equal to $0$ .

$x^{2} + 9 = 0$

Step 7.2.3.2.2

Solve for $x$ .

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Step 7.2.3.2.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 7.2.3.2.2.2

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

$x = \pm \sqrt{- 9}$

Step 7.2.3.2.2.3

Simplify $\pm \sqrt{- 9}$ .

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Step 7.2.3.2.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 7.2.3.2.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 7.2.3.2.2.3.3

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

$x = \pm i \cdot \sqrt{9}$

Step 7.2.3.2.2.3.4

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

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

Step 7.2.3.2.2.3.5

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

$x = \pm i \cdot 3$

Step 7.2.3.2.2.3.6

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

$x = \pm 3 i$

Step 7.2.3.2.2.4

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

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Step 7.2.3.2.2.4.1

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

$x = 3 i$

Step 7.2.3.2.2.4.2

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

$x = - 3 i$

Step 7.2.3.2.2.4.3

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

$x = 3 i, - 3 i$

Step 7.2.4

Set ${(x + 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 7.2.4.1

Set ${(x + 3)}^{2}$ equal to $0$ .

${(x + 3)}^{2} = 0$

Step 7.2.4.2

Solve ${(x + 3)}^{2} = 0$ for $x$ .

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Step 7.2.4.2.1

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 7.2.4.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7.2.5

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 7.2.5.1

Set ${(x - 3)}^{2}$ equal to $0$ .

${(x - 3)}^{2} = 0$

Step 7.2.5.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 7.2.5.2.1

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

$x - 3 = 0$

Step 7.2.5.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 7.2.6

The final solution is all the values that make ${(x^{2} + 9)}^{2} {(x + 3)}^{2} {(x - 3)}^{2} = 0$ true.

$x = 3 i, - 3 i, - 3, 3$

Step 7.3

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

$x = - 3, x = 3$

Step 8

Critical points to evaluate.

$x = 0$

Step 9

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

$\frac{324 {(0)}^{2} (5 {(0)}^{4} + 243)}{{((0) + 3)}^{3} {({(0)}^{2} + 9)}^{3} {((0) - 3)}^{3}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$\frac{324 \cdot 0^{2} (5 \cdot 0 + 243)}{{(0 + 3)}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.1.2

Multiply $5$ by $0$ .

$\frac{324 \cdot 0^{2} (0 + 243)}{{(0 + 3)}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.1.3

Add $0$ and $243$ .

$\frac{324 \cdot 0^{2} \cdot 243}{{(0 + 3)}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.1.4

Multiply $243$ by $324$ .

$\frac{78732 \cdot 0^{2}}{{(0 + 3)}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.1.5

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

$\frac{78732 \cdot 0}{{(0 + 3)}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Rewrite $0$ as $- 1 (0)$ .

$\frac{78732 \cdot 0}{{(- 1 (0) + 3)}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.2.2

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

$\frac{78732 \cdot 0}{{(- 1 \cdot 0 - 1 \cdot - 3)}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.2.3

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot - 3$ .

$\frac{78732 \cdot 0}{{(- 1 \cdot (0 - 3))}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.2.4

Apply the product rule to $- 1 \cdot (0 - 3)$ .

$\frac{78732 \cdot 0}{{(- 1)}^{3} \cdot {(0 - 3)}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.2.5

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

$\frac{78732 \cdot 0}{- 1 \cdot {(0 - 3)}^{3} {(0^{2} + 9)}^{3} {(0 - 3)}^{3}}$

Step 10.2.6

Multiply ${(0 - 3)}^{3}$ by ${(0 - 3)}^{3}$ by adding the exponents.

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Step 10.2.6.1

Move ${(0 - 3)}^{3}$ .

$\frac{78732 \cdot 0}{- 1 \cdot ({(0 - 3)}^{3} {(0 - 3)}^{3}) {(0^{2} + 9)}^{3}}$

Step 10.2.6.2

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

$\frac{78732 \cdot 0}{- 1 \cdot {(0 - 3)}^{3 + 3} {(0^{2} + 9)}^{3}}$

Step 10.2.6.3

Add $3$ and $3$ .

$\frac{78732 \cdot 0}{- 1 \cdot {(0 - 3)}^{6} {(0^{2} + 9)}^{3}}$

Step 10.3

Multiply $78732$ by $0$ .

$\frac{0}{- 1 \cdot {(0 - 3)}^{6} {(0^{2} + 9)}^{3}}$

Step 10.4

Simplify the denominator.

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Step 10.4.1

Subtract $3$ from $0$ .

$\frac{0}{- 1 \cdot {(- 3)}^{6} {(0^{2} + 9)}^{3}}$

Step 10.4.2

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

$\frac{0}{- 1 \cdot {(- 3)}^{6} {(0 + 9)}^{3}}$

Step 10.4.3

Add $0$ and $9$ .

$\frac{0}{- 1 \cdot {(- 3)}^{6} \cdot 9^{3}}$

Step 10.4.4

Combine exponents.

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Step 10.4.4.1

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

$\frac{0}{- 1 \cdot {(- 1 (3))}^{6} \cdot 9^{3}}$

Step 10.4.4.2

Apply the product rule to $- 1 (3)$ .

$\frac{0}{- 1 \cdot ({(- 1)}^{6} \cdot 3^{6}) \cdot 9^{3}}$

Step 10.4.4.3

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

$\frac{0}{- 1 \cdot (1 \cdot 3^{6}) \cdot 9^{3}}$

Step 10.4.4.4

Multiply $3^{6}$ by $1$ .

$\frac{0}{- 1 \cdot 3^{6} \cdot 9^{3}}$

Step 10.4.4.5

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

$\frac{0}{- 1 \cdot ({(3^{2})}^{3} \cdot 3^{6})}$

Step 10.4.4.6

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

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Step 10.4.4.6.1

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

$\frac{0}{- 1 \cdot (3^{2 \cdot 3} \cdot 3^{6})}$

Step 10.4.4.6.2

Multiply $2$ by $3$ .

$\frac{0}{- 1 \cdot (3^{6} \cdot 3^{6})}$

Step 10.4.4.7

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

$\frac{0}{- 1 \cdot 3^{6 + 6}}$

Step 10.4.4.8

Add $6$ and $6$ .

$\frac{0}{- 1 \cdot 3^{12}}$

Step 10.4.5

Raise $3$ to the power of $12$ .

$\frac{0}{- 1 \cdot 531441}$

Step 10.5

Simplify the expression.

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Step 10.5.1

Multiply $- 1$ by $531441$ .

$\frac{0}{- 531441}$

Step 10.5.2

Divide $0$ by $- 531441$ .

$0$

Step 11

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

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Step 11.1

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

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

Step 11.2

Substitute any number, such as $- 6$ , from the interval $(- \infty, 0)$ in the first derivative $- \frac{324 x^{3}}{{(x^{4} - 81)}^{2}}$ to check if the result is negative or positive.

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Step 11.2.1

Replace the variable $x$ with $- 6$ in the expression.

$f' (- 6) = - \frac{324 {(- 6)}^{3}}{{({(- 6)}^{4} - 81)}^{2}}$

Step 11.2.2

Simplify the result.

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Step 11.2.2.1

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

$f' (- 6) = - \frac{324 \cdot - 216}{{({(- 6)}^{4} - 81)}^{2}}$

Step 11.2.2.2

Simplify the denominator.

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Step 11.2.2.2.1

Raise $- 6$ to the power of $4$ .

$f' (- 6) = - \frac{324 \cdot - 216}{{(1296 - 81)}^{2}}$

Step 11.2.2.2.2

Subtract $81$ from $1296$ .

$f' (- 6) = - \frac{324 \cdot - 216}{1215^{2}}$

Step 11.2.2.2.3

Raise $1215$ to the power of $2$ .

$f' (- 6) = - \frac{324 \cdot - 216}{1476225}$

Step 11.2.2.3

Reduce the expression by cancelling the common factors.

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Step 11.2.2.3.1

Multiply $324$ by $- 216$ .

$f' (- 6) = - \frac{- 69984}{1476225}$

Step 11.2.2.3.2

Cancel the common factor of $- 69984$ and $1476225$ .

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Step 11.2.2.3.2.1

Factor $2187$ out of $- 69984$ .

$f' (- 6) = - \frac{2187 (- 32)}{1476225}$

Step 11.2.2.3.2.2

Cancel the common factors.

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Step 11.2.2.3.2.2.1

Factor $2187$ out of $1476225$ .

$f' (- 6) = - \frac{2187 \cdot - 32}{2187 \cdot 675}$

Step 11.2.2.3.2.2.2

Cancel the common factor.

$f' (- 6) = - \frac{2187 \cdot - 32}{2187 \cdot 675}$

Step 11.2.2.3.2.2.3

Rewrite the expression.

$f' (- 6) = - \frac{- 32}{675}$

Step 11.2.2.3.3

Move the negative in front of the fraction.

$f' (- 6) = \frac{32}{675}$

Step 11.2.2.4

The final answer is $\frac{32}{675}$ .

$\frac{32}{675}$

Step 11.3

Substitute any number, such as $2$ , from the interval $(0, \infty)$ in the first derivative $- \frac{324 x^{3}}{{(x^{4} - 81)}^{2}}$ to check if the result is negative or positive.

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Step 11.3.1

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

$f' (2) = - \frac{324 {(2)}^{3}}{{({(2)}^{4} - 81)}^{2}}$

Step 11.3.2

Simplify the result.

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Step 11.3.2.1

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

$f' (2) = - \frac{324 \cdot 8}{{(2^{4} - 81)}^{2}}$

Step 11.3.2.2

Simplify the denominator.

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Step 11.3.2.2.1

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

$f' (2) = - \frac{324 \cdot 8}{{(16 - 81)}^{2}}$

Step 11.3.2.2.2

Subtract $81$ from $16$ .

$f' (2) = - \frac{324 \cdot 8}{{(- 65)}^{2}}$

Step 11.3.2.2.3

Raise $- 65$ to the power of $2$ .

$f' (2) = - \frac{324 \cdot 8}{4225}$

Step 11.3.2.3

Multiply $324$ by $8$ .

$f' (2) = - \frac{2592}{4225}$

Step 11.3.2.4

The final answer is $- \frac{2592}{4225}$ .

$- \frac{2592}{4225}$

Step 11.4

Since the first derivative changed signs from positive to negative around $x = 0$ , then $x = 0$ is a local maximum.

$x = 0$ is a local maximum

Step 12