Calculus Examples

$y = 3 \arccos (x^{6})$

Step 1

Write $y = 3 \arccos (x^{6})$ as a function.

$f (x) = 3 \arccos (x^{6})$

Step 2

Find the first derivative of the function.

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Step 2.1

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

$3 \frac{d}{d x} [\arccos (x^{6})]$

Step 2.2

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

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Step 2.2.1

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

$3 (\frac{d}{d u} [\arccos (u)] \frac{d}{d x} [x^{6}])$

Step 2.2.2

The derivative of $\arccos (u)$ with respect to $u$ is $- \frac{1}{\sqrt{1 - u^{2}}}$ .

$3 (- \frac{1}{\sqrt{1 - u^{2}}} \frac{d}{d x} [x^{6}])$

Step 2.2.3

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

$3 (- \frac{1}{\sqrt{1 - {(x^{6})}^{2}}} \frac{d}{d x} [x^{6}])$

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

Multiply the exponents in ${(x^{6})}^{2}$ .

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Step 2.3.1.1

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

$3 (- \frac{1}{\sqrt{1 - x^{6 \cdot 2}}} \frac{d}{d x} [x^{6}])$

Step 2.3.1.2

Multiply $6$ by $2$ .

$3 (- \frac{1}{\sqrt{1 - x^{12}}} \frac{d}{d x} [x^{6}])$

Step 2.3.2

Combine fractions.

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Step 2.3.2.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{1}{\sqrt{1 - x^{12}}} \frac{d}{d x} [x^{6}])$

Step 2.3.2.2

Combine $\frac{1}{\sqrt{1 - x^{12}}}$ and $- 3$ .

$\frac{- 3}{\sqrt{1 - x^{12}}} \frac{d}{d x} [x^{6}]$

Step 2.3.2.3

Move the negative in front of the fraction.

$- \frac{3}{\sqrt{1 - x^{12}}} \frac{d}{d x} [x^{6}]$

Step 2.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$- \frac{3}{\sqrt{1 - x^{12}}} (6 x^{5})$

Step 2.3.4

Combine fractions.

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Step 2.3.4.1

Multiply $6$ by $- 1$ .

$- 6 \frac{3}{\sqrt{1 - x^{12}}} x^{5}$

Step 2.3.4.2

Combine $- 6$ and $\frac{3}{\sqrt{1 - x^{12}}}$ .

$\frac{- 6 \cdot 3}{\sqrt{1 - x^{12}}} x^{5}$

Step 2.3.4.3

Multiply $- 6$ by $3$ .

$\frac{- 18}{\sqrt{1 - x^{12}}} x^{5}$

Step 2.3.4.4

Combine $\frac{- 18}{\sqrt{1 - x^{12}}}$ and $x^{5}$ .

$\frac{- 18 x^{5}}{\sqrt{1 - x^{12}}}$

Step 2.3.4.5

Move the negative in front of the fraction.

$- \frac{18 x^{5}}{\sqrt{1 - x^{12}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - x^{12}}$ as ${(1 - x^{12})}^{\frac{1}{2}}$ .

$f'' (x) = \frac{d}{d x} (- \frac{18 x^{5}}{{(1 - x^{12})}^{\frac{1}{2}}})$

Step 3.1.2

Since $- 18$ is constant with respect to $x$ , the derivative of $- \frac{18 x^{5}}{{(1 - x^{12})}^{\frac{1}{2}}}$ with respect to $x$ is $- 18 \frac{d}{d x} [\frac{x^{5}}{{(1 - x^{12})}^{\frac{1}{2}}}]$ .

$f'' (x) = - 18 \frac{d}{d x} \frac{x^{5}}{{(1 - x^{12})}^{\frac{1}{2}}}$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{5}$ and $g (x) = {(1 - x^{12})}^{\frac{1}{2}}$ .

$f'' (x) = - 18 \frac{{(1 - x^{12})}^{\frac{1}{2}} \frac{d}{d x} (x^{5}) - x^{5} \frac{d}{d x} {(1 - x^{12})}^{\frac{1}{2}}}{{({(1 - x^{12})}^{\frac{1}{2}})}^{2}}$

Step 3.3

Multiply the exponents in ${({(1 - x^{12})}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1

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

$f'' (x) = - 18 \frac{{(1 - x^{12})}^{\frac{1}{2}} \frac{d}{d x} (x^{5}) - x^{5} \frac{d}{d x} {(1 - x^{12})}^{\frac{1}{2}}}{{(1 - x^{12})}^{\frac{1}{2} \cdot 2}}$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Cancel the common factor.

$f'' (x) = - 18 \frac{{(1 - x^{12})}^{\frac{1}{2}} \frac{d}{d x} (x^{5}) - x^{5} \frac{d}{d x} {(1 - x^{12})}^{\frac{1}{2}}}{{(1 - x^{12})}^{\frac{1}{2} \cdot 2}}$

Step 3.3.2.2

Rewrite the expression.

$f'' (x) = - 18 \frac{{(1 - x^{12})}^{\frac{1}{2}} \frac{d}{d x} (x^{5}) - x^{5} \frac{d}{d x} {(1 - x^{12})}^{\frac{1}{2}}}{1 - x^{12}}$

Step 3.4

Simplify.

$f'' (x) = - 18 \frac{{(1 - x^{12})}^{\frac{1}{2}} \frac{d}{d x} (x^{5}) - x^{5} \frac{d}{d x} {(1 - x^{12})}^{\frac{1}{2}}}{1 - x^{12}}$

Step 3.5

Differentiate using the Power Rule.

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Step 3.5.1

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

$f'' (x) = - 18 \frac{{(1 - x^{12})}^{\frac{1}{2}} (5 x^{4}) - x^{5} \frac{d}{d x} {(1 - x^{12})}^{\frac{1}{2}}}{1 - x^{12}}$

Step 3.5.2

Move $5$ to the left of ${(1 - x^{12})}^{\frac{1}{2}}$ .

$f'' (x) = - 18 \frac{5 \cdot ({(1 - x^{12})}^{\frac{1}{2}} x^{4}) - x^{5} \frac{d}{d x} {(1 - x^{12})}^{\frac{1}{2}}}{1 - x^{12}}$

Step 3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 1 - x^{12}$ .

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Step 3.6.1

To apply the Chain Rule, set $u$ as $1 - x^{12}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{d}{d u} (u^{\frac{1}{2}}) \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.6.2

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

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.6.3

Replace all occurrences of $u$ with $1 - x^{12}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{1}{2} \cdot {(1 - x^{12})}^{\frac{1}{2} - 1} \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{1}{2} \cdot {(1 - x^{12})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.8

Combine $- 1$ and $\frac{2}{2}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{1}{2} \cdot {(1 - x^{12})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.9

Combine the numerators over the common denominator.

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{1}{2} \cdot {(1 - x^{12})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $2$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{1}{2} \cdot {(1 - x^{12})}^{\frac{1 - 2}{2}} \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.10.2

Subtract $2$ from $1$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{1}{2} \cdot {(1 - x^{12})}^{\frac{- 1}{2}} \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.11

Combine fractions.

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Step 3.11.1

Move the negative in front of the fraction.

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{1}{2} \cdot {(1 - x^{12})}^{- \frac{1}{2}} \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.11.2

Combine $\frac{1}{2}$ and ${(1 - x^{12})}^{- \frac{1}{2}}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{{(1 - x^{12})}^{- \frac{1}{2}}}{2} \cdot \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.11.3

Move ${(1 - x^{12})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - x^{5} (\frac{1}{2 {(1 - x^{12})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (1 - x^{12}))}{1 - x^{12}}$

Step 3.11.4

Combine $\frac{1}{2 {(1 - x^{12})}^{\frac{1}{2}}}$ and $x^{5}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - \frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (1 - x^{12})}{1 - x^{12}}$

Step 3.12

By the Sum Rule, the derivative of $1 - x^{12}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{12}]$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - \frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}} \cdot (\frac{d}{d x} (1) + \frac{d}{d x} (- x^{12}))}{1 - x^{12}}$

Step 3.13

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - \frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}} \cdot (0 + \frac{d}{d x} (- x^{12}))}{1 - x^{12}}$

Step 3.14

Add $0$ and $\frac{d}{d x} [- x^{12}]$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - \frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (- x^{12})}{1 - x^{12}}$

Step 3.15

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{12}$ with respect to $x$ is $- \frac{d}{d x} [x^{12}]$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} - \frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}} \cdot (- \frac{d}{d x} x^{12})}{1 - x^{12}}$

Step 3.16

Multiply.

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Step 3.16.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + 1 (\frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}}) \cdot \frac{d}{d x} (x^{12})}{1 - x^{12}}$

Step 3.16.2

Multiply $\frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}}$ by $1$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}} \cdot \frac{d}{d x} (x^{12})}{1 - x^{12}}$

Step 3.17

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 12$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}} \cdot (12 x^{11})}{1 - x^{12}}$

Step 3.18

Combine fractions.

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Step 3.18.1

Combine $12$ and $\frac{x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{12 x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}} \cdot x^{11}}{1 - x^{12}}$

Step 3.18.2

Combine $\frac{12 x^{5}}{2 {(1 - x^{12})}^{\frac{1}{2}}}$ and $x^{11}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{12 x^{5} x^{11}}{2 {(1 - x^{12})}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.19

Multiply $x^{5}$ by $x^{11}$ by adding the exponents.

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Step 3.19.1

Move $x^{11}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{12 (x^{11} x^{5})}{2 {(1 - x^{12})}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.19.2

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

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{12 x^{11 + 5}}{2 {(1 - x^{12})}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.19.3

Add $11$ and $5$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{12 x^{16}}{2 {(1 - x^{12})}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.20

Factor $2$ out of $12 x^{16}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{2 (6 x^{16})}{2 {(1 - x^{12})}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.21

Cancel the common factors.

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Step 3.21.1

Factor $2$ out of $2 {(1 - x^{12})}^{\frac{1}{2}}$ .

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{2 (6 x^{16})}{2 ({(1 - x^{12})}^{\frac{1}{2}})}}{1 - x^{12}}$

Step 3.21.2

Cancel the common factor.

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{2 (6 x^{16})}{2 {(1 - x^{12})}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.21.3

Rewrite the expression.

$f'' (x) = - 18 \frac{5 {(1 - x^{12})}^{\frac{1}{2}} x^{4} + \frac{6 x^{16}}{{(1 - x^{12})}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.22

Reorder $1$ and $- x^{12}$ .

$f'' (x) = - 18 \frac{5 x^{4} {(- x^{12} + 1)}^{\frac{1}{2}} + \frac{6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.23

To write $5 x^{4} {(- x^{12} + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(- x^{12} + 1)}^{\frac{1}{2}}}{{(- x^{12} + 1)}^{\frac{1}{2}}}$ .

$f'' (x) = - 18 \frac{\frac{5 x^{4} {(- x^{12} + 1)}^{\frac{1}{2}} {(- x^{12} + 1)}^{\frac{1}{2}}}{{(- x^{12} + 1)}^{\frac{1}{2}}} + \frac{6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.24

Combine the numerators over the common denominator.

$f'' (x) = - 18 \frac{\frac{5 x^{4} {(- x^{12} + 1)}^{\frac{1}{2}} {(- x^{12} + 1)}^{\frac{1}{2}} + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.25

Multiply ${(- x^{12} + 1)}^{\frac{1}{2}}$ by ${(- x^{12} + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 3.25.1

Move ${(- x^{12} + 1)}^{\frac{1}{2}}$ .

$f'' (x) = - 18 \frac{\frac{5 x^{4} ({(- x^{12} + 1)}^{\frac{1}{2}} {(- x^{12} + 1)}^{\frac{1}{2}}) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.25.2

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

$f'' (x) = - 18 \frac{\frac{5 x^{4} {(- x^{12} + 1)}^{\frac{1}{2} + \frac{1}{2}} + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.25.3

Combine the numerators over the common denominator.

$f'' (x) = - 18 \frac{\frac{5 x^{4} {(- x^{12} + 1)}^{\frac{1 + 1}{2}} + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.25.4

Add $1$ and $1$ .

$f'' (x) = - 18 \frac{\frac{5 x^{4} {(- x^{12} + 1)}^{\frac{2}{2}} + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.25.5

Divide $2$ by $2$ .

$f'' (x) = - 18 \frac{\frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.26

Simplify $5 x^{4} {(- x^{12} + 1)}^{1}$ .

$f'' (x) = - 18 \frac{\frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$

Step 3.27

Rewrite $\frac{\frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}}{1 - x^{12}}$ as a product.

$f'' (x) = - 18 (\frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}} \cdot \frac{1}{1 - x^{12}})$

Step 3.28

Multiply $\frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}}}$ by $\frac{1}{1 - x^{12}}$ .

$f'' (x) = - 18 \frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}} (1 - x^{12})}$

Step 3.29

Reorder terms.

$f'' (x) = - 18 \frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}} (- x^{12} + 1)}$

Step 3.30

Multiply ${(- x^{12} + 1)}^{\frac{1}{2}}$ by $- x^{12} + 1$ by adding the exponents.

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Step 3.30.1

Multiply ${(- x^{12} + 1)}^{\frac{1}{2}}$ by $- x^{12} + 1$ .

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Step 3.30.1.1

Raise $- x^{12} + 1$ to the power of $1$ .

$f'' (x) = - 18 \frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2}} (- x^{12} + 1)}$

Step 3.30.1.2

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

$f'' (x) = - 18 \frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2} + 1}}$

Step 3.30.2

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

$f'' (x) = - 18 \frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1}{2} + \frac{2}{2}}}$

Step 3.30.3

Combine the numerators over the common denominator.

$f'' (x) = - 18 \frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{1 + 2}{2}}}$

Step 3.30.4

Add $1$ and $2$ .

$f'' (x) = - 18 \frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.31

Combine $- 18$ and $\frac{5 x^{4} (- x^{12} + 1) + 6 x^{16}}{{(- x^{12} + 1)}^{\frac{3}{2}}}$ .

$f'' (x) = \frac{- 18 (5 x^{4} (- x^{12} + 1) + 6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.32

Move the negative in front of the fraction.

$f'' (x) = - \frac{18 (5 x^{4} (- x^{12} + 1) + 6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33

Simplify.

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Step 3.33.1

Apply the distributive property.

$f'' (x) = - \frac{18 (5 x^{4} (- x^{12}) + 5 x^{4} \cdot 1 + 6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.2

Apply the distributive property.

$f'' (x) = - \frac{18 (5 x^{4} (- x^{12})) + 18 (5 x^{4} \cdot 1) + 18 (6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.3

Simplify the numerator.

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Step 3.33.3.1

Simplify each term.

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Step 3.33.3.1.1

Multiply $x^{4}$ by $x^{12}$ by adding the exponents.

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Step 3.33.3.1.1.1

Move $x^{12}$ .

$f'' (x) = - \frac{18 (5 (x^{12} x^{4}) \cdot - 1) + 18 (5 x^{4} \cdot 1) + 18 (6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.3.1.1.2

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

$f'' (x) = - \frac{18 (5 x^{12 + 4} \cdot - 1) + 18 (5 x^{4} \cdot 1) + 18 (6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.3.1.1.3

Add $12$ and $4$ .

$f'' (x) = - \frac{18 (5 x^{16} \cdot - 1) + 18 (5 x^{4} \cdot 1) + 18 (6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.3.1.2

Multiply $- 1$ by $5$ .

$f'' (x) = - \frac{18 (- 5 x^{16}) + 18 (5 x^{4} \cdot 1) + 18 (6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.3.1.3

Multiply $- 5$ by $18$ .

$f'' (x) = - \frac{- 90 x^{16} + 18 (5 x^{4} \cdot 1) + 18 (6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.3.1.4

Multiply $5$ by $1$ .

$f'' (x) = - \frac{- 90 x^{16} + 18 (5 x^{4}) + 18 (6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.3.1.5

Multiply $5$ by $18$ .

$f'' (x) = - \frac{- 90 x^{16} + 90 x^{4} + 18 (6 x^{16})}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.3.1.6

Multiply $6$ by $18$ .

$f'' (x) = - \frac{- 90 x^{16} + 90 x^{4} + 108 x^{16}}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.3.2

Add $- 90 x^{16}$ and $108 x^{16}$ .

$f'' (x) = - \frac{18 x^{16} + 90 x^{4}}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.4

Factor $18 x^{4}$ out of $18 x^{16} + 90 x^{4}$ .

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Step 3.33.4.1

Factor $18 x^{4}$ out of $18 x^{16}$ .

$f'' (x) = - \frac{18 x^{4} (x^{12}) + 90 x^{4}}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.4.2

Factor $18 x^{4}$ out of $90 x^{4}$ .

$f'' (x) = - \frac{18 x^{4} (x^{12}) + 18 x^{4} (5)}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 3.33.4.3

Factor $18 x^{4}$ out of $18 x^{4} (x^{12}) + 18 x^{4} (5)$ .

$f'' (x) = - \frac{18 x^{4} (x^{12} + 5)}{{(- x^{12} + 1)}^{\frac{3}{2}}}$

Step 4

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

$- \frac{18 x^{5}}{\sqrt{1 - x^{12}}} = 0$

Step 5

Set the numerator equal to zero.

$18 x^{5} = 0$

Step 6

Solve the equation for $x$ .

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Step 6.1

Divide each term in $18 x^{5} = 0$ by $18$ and simplify.

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Step 6.1.1

Divide each term in $18 x^{5} = 0$ by $18$ .

$\frac{18 x^{5}}{18} = \frac{0}{18}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $18$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{18 x^{5}}{18} = \frac{0}{18}$

Step 6.1.2.1.2

Divide $x^{5}$ by $1$ .

$x^{5} = \frac{0}{18}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Divide $0$ by $18$ .

$x^{5} = 0$

Step 6.2

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

$x = \sqrt[5]{0}$

Step 6.3

Simplify $\sqrt[5]{0}$ .

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Step 6.3.1

Rewrite $0$ as $0^{5}$ .

$x = \sqrt[5]{0^{5}}$

Step 6.3.2

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

$x = 0$

Step 7

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.

$- \frac{18 {(0)}^{4} ({(0)}^{12} + 5)}{{(- {(0)}^{12} + 1)}^{\frac{3}{2}}}$

Step 8

Evaluate the second derivative.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$- \frac{18 \cdot 0^{4} (0 + 5)}{{(- 0^{12} + 1)}^{\frac{3}{2}}}$

Step 8.1.2

Add $0$ and $5$ .

$- \frac{18 \cdot 0^{4} \cdot 5}{{(- 0^{12} + 1)}^{\frac{3}{2}}}$

Step 8.1.3

Multiply $5$ by $18$ .

$- \frac{90 \cdot 0^{4}}{{(- 0^{12} + 1)}^{\frac{3}{2}}}$

Step 8.1.4

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

$- \frac{90 \cdot 0}{{(- 0^{12} + 1)}^{\frac{3}{2}}}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

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

$- \frac{90 \cdot 0}{{(- 0 + 1)}^{\frac{3}{2}}}$

Step 8.2.1.2

Multiply $- 1$ by $0$ .

$- \frac{90 \cdot 0}{{(0 + 1)}^{\frac{3}{2}}}$

Step 8.2.2

Add $0$ and $1$ .

$- \frac{90 \cdot 0}{1^{\frac{3}{2}}}$

Step 8.2.3

One to any power is one.

$- \frac{90 \cdot 0}{1}$

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $90$ by $0$ .

$- \frac{0}{1}$

Step 8.3.2

Divide $0$ by $1$ .

$- 0$

Step 8.3.3

Multiply $- 1$ by $0$ .

$0$

Step 9

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

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Step 9.1

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

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

Step 9.2

Substitute any number, such as $- 0.5$ , from the interval $(- \infty, 0)$ in the first derivative $- \frac{18 x^{5}}{\sqrt{1 - x^{12}}}$ to check if the result is negative or positive.

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Step 9.2.1

Replace the variable $x$ with $- 0.5$ in the expression.

$f' (- 0.5) = - \frac{18 {(- 0.5)}^{5}}{\sqrt{1 - {(- 0.5)}^{12}}}$

Step 9.2.2

Simplify the result.

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Step 9.2.2.1

Raise $- 0.5$ to the power of $5$ .

$f' (- 0.5) = - \frac{18 \cdot - 0.03125}{\sqrt{1 - {(- 0.5)}^{12}}}$

Step 9.2.2.2

Simplify the denominator.

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Step 9.2.2.2.1

Raise $- 0.5$ to the power of $12$ .

$f' (- 0.5) = - \frac{18 \cdot - 0.03125}{\sqrt{1 - 1 \cdot 0.00024414}}$

Step 9.2.2.2.2

Multiply $- 1$ by $0.00024414$ .

$f' (- 0.5) = - \frac{18 \cdot - 0.03125}{\sqrt{1 - 0.00024414}}$

Step 9.2.2.2.3

Subtract $0.00024414$ from $1$ .

$f' (- 0.5) = - \frac{18 \cdot - 0.03125}{\sqrt{0.99975585}}$

Step 9.2.2.3

Simplify the expression.

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Step 9.2.2.3.1

Multiply $18$ by $- 0.03125$ .

$f' (- 0.5) = - \frac{- 0.5625}{\sqrt{0.99975585}}$

Step 9.2.2.3.2

Move the negative in front of the fraction.

$f' (- 0.5) = \frac{0.5625}{\sqrt{0.99975585}}$

Step 9.2.2.4

Multiply $\frac{0.5625}{\sqrt{0.99975585}}$ by $\frac{\sqrt{0.99975585}}{\sqrt{0.99975585}}$ .

$f' (- 0.5) = \frac{0.5625}{\sqrt{0.99975585}} \cdot \frac{\sqrt{0.99975585}}{\sqrt{0.99975585}}$

Step 9.2.2.5

Combine and simplify the denominator.

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Step 9.2.2.5.1

Multiply $\frac{0.5625}{\sqrt{0.99975585}}$ by $\frac{\sqrt{0.99975585}}{\sqrt{0.99975585}}$ .

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{\sqrt{0.99975585} \sqrt{0.99975585}}$

Step 9.2.2.5.2

Raise $\sqrt{0.99975585}$ to the power of $1$ .

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{\sqrt{0.99975585} \sqrt{0.99975585}}$

Step 9.2.2.5.3

Raise $\sqrt{0.99975585}$ to the power of $1$ .

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{\sqrt{0.99975585} \sqrt{0.99975585}}$

Step 9.2.2.5.4

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

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{{\sqrt{0.99975585}}^{1 + 1}}$

Step 9.2.2.5.5

Add $1$ and $1$ .

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{{\sqrt{0.99975585}}^{2}}$

Step 9.2.2.5.6

Rewrite ${\sqrt{0.99975585}}^{2}$ as $0.99975585$ .

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Step 9.2.2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{0.99975585}$ as ${0.99975585}^{\frac{1}{2}}$ .

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{{({0.99975585}^{\frac{1}{2}})}^{2}}$

Step 9.2.2.5.6.2

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

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{{0.99975585}^{\frac{1}{2} \cdot 2}}$

Step 9.2.2.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{{0.99975585}^{\frac{2}{2}}}$

Step 9.2.2.5.6.4

Cancel the common factor of $2$ .

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Step 9.2.2.5.6.4.1

Cancel the common factor.

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{{0.99975585}^{\frac{2}{2}}}$

Step 9.2.2.5.6.4.2

Rewrite the expression.

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{0.99975585}$

Step 9.2.2.5.6.5

Evaluate the exponent.

$f' (- 0.5) = \frac{0.5625 \sqrt{0.99975585}}{0.99975585}$

Step 9.2.2.6

Multiply $0.5625$ by $\sqrt{0.99975585}$ .

$f' (- 0.5) = \frac{0.56243133}{0.99975585}$

Step 9.2.2.7

Divide $0.56243133$ by $0.99975585$ .

$f' (- 0.5) = - (- 1 \cdot 0.56256867)$

Step 9.2.2.8

Multiply $- (- 1 \cdot 0.56256867)$ .

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Step 9.2.2.8.1

Multiply $- 1$ by $0.56256867$ .

$f' (- 0.5) = 0.56256867$

Step 9.2.2.8.2

Multiply $- 1$ by $- 0.56256867$ .

$f' (- 0.5) = 0.56256867$

Step 9.2.2.9

The final answer is $0.56256867$ .

$0.56256867$

Step 9.3

Substitute any number, such as $0.5$ , from the interval $(0, \infty)$ in the first derivative $- \frac{18 x^{5}}{\sqrt{1 - x^{12}}}$ to check if the result is negative or positive.

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Step 9.3.1

Replace the variable $x$ with $0.5$ in the expression.

$f' (0.5) = - \frac{18 {(0.5)}^{5}}{\sqrt{1 - {(0.5)}^{12}}}$

Step 9.3.2

Simplify the result.

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Step 9.3.2.1

Raise $0.5$ to the power of $5$ .

$f' (0.5) = - \frac{18 \cdot 0.03125}{\sqrt{1 - {0.5}^{12}}}$

Step 9.3.2.2

Simplify the denominator.

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Step 9.3.2.2.1

Raise $0.5$ to the power of $12$ .

$f' (0.5) = - \frac{18 \cdot 0.03125}{\sqrt{1 - 1 \cdot 0.00024414}}$

Step 9.3.2.2.2

Multiply $- 1$ by $0.00024414$ .

$f' (0.5) = - \frac{18 \cdot 0.03125}{\sqrt{1 - 0.00024414}}$

Step 9.3.2.2.3

Subtract $0.00024414$ from $1$ .

$f' (0.5) = - \frac{18 \cdot 0.03125}{\sqrt{0.99975585}}$

Step 9.3.2.3

Multiply $18$ by $0.03125$ .

$f' (0.5) = - \frac{0.5625}{\sqrt{0.99975585}}$

Step 9.3.2.4

Multiply $\frac{0.5625}{\sqrt{0.99975585}}$ by $\frac{\sqrt{0.99975585}}{\sqrt{0.99975585}}$ .

$f' (0.5) = - (\frac{0.5625}{\sqrt{0.99975585}} \cdot \frac{\sqrt{0.99975585}}{\sqrt{0.99975585}})$

Step 9.3.2.5

Combine and simplify the denominator.

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Step 9.3.2.5.1

Multiply $\frac{0.5625}{\sqrt{0.99975585}}$ by $\frac{\sqrt{0.99975585}}{\sqrt{0.99975585}}$ .

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{\sqrt{0.99975585} \sqrt{0.99975585}}$

Step 9.3.2.5.2

Raise $\sqrt{0.99975585}$ to the power of $1$ .

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{\sqrt{0.99975585} \sqrt{0.99975585}}$

Step 9.3.2.5.3

Raise $\sqrt{0.99975585}$ to the power of $1$ .

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{\sqrt{0.99975585} \sqrt{0.99975585}}$

Step 9.3.2.5.4

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

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{{\sqrt{0.99975585}}^{1 + 1}}$

Step 9.3.2.5.5

Add $1$ and $1$ .

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{{\sqrt{0.99975585}}^{2}}$

Step 9.3.2.5.6

Rewrite ${\sqrt{0.99975585}}^{2}$ as $0.99975585$ .

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Step 9.3.2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{0.99975585}$ as ${0.99975585}^{\frac{1}{2}}$ .

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{{({0.99975585}^{\frac{1}{2}})}^{2}}$

Step 9.3.2.5.6.2

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

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{{0.99975585}^{\frac{1}{2} \cdot 2}}$

Step 9.3.2.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{{0.99975585}^{\frac{2}{2}}}$

Step 9.3.2.5.6.4

Cancel the common factor of $2$ .

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Step 9.3.2.5.6.4.1

Cancel the common factor.

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{{0.99975585}^{\frac{2}{2}}}$

Step 9.3.2.5.6.4.2

Rewrite the expression.

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{0.99975585}$

Step 9.3.2.5.6.5

Evaluate the exponent.

$f' (0.5) = - \frac{0.5625 \sqrt{0.99975585}}{0.99975585}$

Step 9.3.2.6

Multiply $0.5625$ by $\sqrt{0.99975585}$ .

$f' (0.5) = - \frac{0.56243133}{0.99975585}$

Step 9.3.2.7

Simplify the expression.

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Step 9.3.2.7.1

Divide $0.56243133$ by $0.99975585$ .

$f' (0.5) = - 1 \cdot 0.56256867$

Step 9.3.2.7.2

Multiply $- 1$ by $0.56256867$ .

$f' (0.5) = - 0.56256867$

Step 9.3.2.8

The final answer is $- 0.56256867$ .

$- 0.56256867$

Step 9.4

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 10