Calculus Examples

$y = \frac{13 x}{1 + 0.25 x^{2}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Since $13$ is constant with respect to $x$ , the derivative of $\frac{13 x}{1 + 0.25 x^{2}}$ with respect to $x$ is $13 \frac{d}{d x} [\frac{x}{1 + 0.25 x^{2}}]$ .

$13 \frac{d}{d x} [\frac{x}{1 + 0.25 x^{2}}]$

Step 1.2

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 $f (x) = x$ and $g (x) = 1 + 0.25 x^{2}$ .

$13 \frac{(1 + 0.25 x^{2}) \frac{d}{d x} [x] - x \frac{d}{d x} [1 + 0.25 x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

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

$13 \frac{(1 + 0.25 x^{2}) \cdot 1 - x \frac{d}{d x} [1 + 0.25 x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.3.2

Multiply $1 + 0.25 x^{2}$ by $1$ .

$13 \frac{1 + 0.25 x^{2} - x \frac{d}{d x} [1 + 0.25 x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.3.3

By the Sum Rule, the derivative of $1 + 0.25 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [0.25 x^{2}]$ .

$13 \frac{1 + 0.25 x^{2} - x (\frac{d}{d x} [1] + \frac{d}{d x} [0.25 x^{2}])}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.3.4

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

$13 \frac{1 + 0.25 x^{2} - x (0 + \frac{d}{d x} [0.25 x^{2}])}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.3.5

Add $0$ and $\frac{d}{d x} [0.25 x^{2}]$ .

$13 \frac{1 + 0.25 x^{2} - x \frac{d}{d x} [0.25 x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.3.6

Since $0.25$ is constant with respect to $x$ , the derivative of $0.25 x^{2}$ with respect to $x$ is $0.25 \frac{d}{d x} [x^{2}]$ .

$13 \frac{1 + 0.25 x^{2} - x (0.25 \frac{d}{d x} [x^{2}])}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.3.7

Multiply $0.25$ by $- 1$ .

$13 \frac{1 + 0.25 x^{2} - 0.25 x \frac{d}{d x} [x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.3.8

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

$13 \frac{1 + 0.25 x^{2} - 0.25 x (2 x)}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.3.9

Multiply $2$ by $- 0.25$ .

$13 \frac{1 + 0.25 x^{2} - 0.5 x \cdot x}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.4

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

$13 \frac{1 + 0.25 x^{2} - 0.5 (x^{1} x)}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.5

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

$13 \frac{1 + 0.25 x^{2} - 0.5 (x^{1} x^{1})}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.6

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

$13 \frac{1 + 0.25 x^{2} - 0.5 x^{1 + 1}}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.7

Add $1$ and $1$ .

$13 \frac{1 + 0.25 x^{2} - 0.5 x^{2}}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.8

Subtract $0.5 x^{2}$ from $0.25 x^{2}$ .

$13 \frac{1 - 0.25 x^{2}}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.9

Combine $13$ and $\frac{1 - 0.25 x^{2}}{{(1 + 0.25 x^{2})}^{2}}$ .

$\frac{13 (1 - 0.25 x^{2})}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

$\frac{13 \cdot 1 + 13 (- 0.25 x^{2})}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.10.2

Simplify each term.

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Step 1.10.2.1

Multiply $13$ by $1$ .

$\frac{13 + 13 (- 0.25 x^{2})}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.10.2.2

Multiply $- 0.25$ by $13$ .

$\frac{13 - 3.25 x^{2}}{{(1 + 0.25 x^{2})}^{2}}$

Step 1.10.3

Reorder terms.

$\frac{- 3.25 x^{2} + 13}{{(0.25 x^{2} + 1)}^{2}}$

Step 1.10.4

Simplify the numerator.

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Step 1.10.4.1

Factor $3.25$ out of $- 3.25 x^{2} + 13$ .

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Step 1.10.4.1.1

Factor $3.25$ out of $- 3.25 x^{2}$ .

$\frac{3.25 (- x^{2}) + 13}{{(0.25 x^{2} + 1)}^{2}}$

Step 1.10.4.1.2

Factor $3.25$ out of $13$ .

$\frac{3.25 (- x^{2}) + 3.25 (4)}{{(0.25 x^{2} + 1)}^{2}}$

Step 1.10.4.1.3

Factor $3.25$ out of $3.25 (- x^{2}) + 3.25 (4)$ .

$\frac{3.25 (- x^{2} + 4)}{{(0.25 x^{2} + 1)}^{2}}$

Step 1.10.4.2

Rewrite $4$ as $2^{2}$ .

$\frac{3.25 (- x^{2} + 2^{2})}{{(0.25 x^{2} + 1)}^{2}}$

Step 1.10.4.3

Reorder $- x^{2}$ and $2^{2}$ .

$\frac{3.25 (2^{2} - x^{2})}{{(0.25 x^{2} + 1)}^{2}}$

Step 1.10.4.4

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

$\frac{3.25 (2 + x) (2 - x)}{{(0.25 x^{2} + 1)}^{2}}$

Step 1.10.5

Simplify the denominator.

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Step 1.10.5.1

Factor $0.25$ out of $0.25 x^{2} + 1$ .

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Step 1.10.5.1.1

Factor $0.25$ out of $0.25 x^{2}$ .

$\frac{3.25 (2 + x) (2 - x)}{{(0.25 (x^{2}) + 1)}^{2}}$

Step 1.10.5.1.2

Factor $0.25$ out of $1$ .

$\frac{3.25 (2 + x) (2 - x)}{{(0.25 x^{2} + 0.25 \cdot 4)}^{2}}$

Step 1.10.5.1.3

Factor $0.25$ out of $0.25 x^{2} + 0.25 \cdot 4$ .

$\frac{3.25 (2 + x) (2 - x)}{{(0.25 (x^{2} + 4))}^{2}}$

Step 1.10.5.2

Apply the product rule to $0.25 (x^{2} + 4)$ .

$\frac{3.25 (2 + x) (2 - x)}{{0.25}^{2} {(x^{2} + 4)}^{2}}$

Step 1.10.5.3

Raise $0.25$ to the power of $2$ .

$\frac{3.25 (2 + x) (2 - x)}{0.0625 {(x^{2} + 4)}^{2}}$

Step 1.10.6

Factor $3.25$ out of $3.25 (2 + x) (2 - x)$ .

$\frac{3.25 ((2 + x) (2 - x))}{0.0625 {(x^{2} + 4)}^{2}}$

Step 1.10.7

Factor $0.0625$ out of $0.0625 {(x^{2} + 4)}^{2}$ .

$\frac{3.25 ((2 + x) (2 - x))}{0.0625 ({(x^{2} + 4)}^{2})}$

Step 1.10.8

Separate fractions.

$\frac{3.25}{0.0625} \cdot \frac{(2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$

Step 1.10.9

Divide $3.25$ by $0.0625$ .

$52 \frac{(2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$

Step 1.10.10

Combine $52$ and $\frac{(2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$ .

$\frac{52 (2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $52$ is constant with respect to $x$ , the derivative of $\frac{52 (2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$ with respect to $x$ is $52 \frac{d}{d x} [\frac{(2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}]$ .

$f'' (x) = 52 \frac{d}{d x} (\frac{(2 + x) (2 - x)}{{(x^{2} + 4)}^{2}})$

Step 2.2

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} \frac{d}{d x} ((2 + x) (2 - x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{({(x^{2} + 4)}^{2})}^{2}})$

Step 2.3

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

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Step 2.3.1

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

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} \frac{d}{d x} ((2 + x) (2 - x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{2 \cdot 2}})$

Step 2.3.2

Multiply $2$ by $2$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} \frac{d}{d x} ((2 + x) (2 - x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 2 + x$ and $g (x) = 2 - x$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} ((2 + x) \frac{d}{d x} (2 - x) + (2 - x) \frac{d}{d x} (2 + x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5

Differentiate.

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Step 2.5.1

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

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} ((2 + x) (\frac{d}{d x} (2) + \frac{d}{d x} (- x)) + (2 - x) \frac{d}{d x} (2 + x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} ((2 + x) (0 + \frac{d}{d x} (- x)) + (2 - x) \frac{d}{d x} (2 + x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.3

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

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} ((2 + x) \frac{d}{d x} (- x) + (2 - x) \frac{d}{d x} (2 + x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.4

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

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} ((2 + x) (- \frac{d}{d x} x) + (2 - x) \frac{d}{d x} (2 + x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.5

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

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} ((2 + x) (- 1 \cdot 1) + (2 - x) \frac{d}{d x} (2 + x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.6

Simplify the expression.

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Step 2.5.6.1

Multiply $- 1$ by $1$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} ((2 + x) \cdot - 1 + (2 - x) \frac{d}{d x} (2 + x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.6.2

Move $- 1$ to the left of $2 + x$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- 1 \cdot (2 + x) + (2 - x) \frac{d}{d x} (2 + x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.6.3

Rewrite $- 1 (2 + x)$ as $- (2 + x)$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + (2 - x) \frac{d}{d x} (2 + x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.7

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

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + (2 - x) (\frac{d}{d x} (2) + \frac{d}{d x} (x))) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.8

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + (2 - x) (0 + \frac{d}{d x} (x))) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.9

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

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + (2 - x) \frac{d}{d x} (x)) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.10

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

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + (2 - x) \cdot 1) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.5.11

Multiply $2 - x$ by $1$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + 2 - x) - (2 + x) (2 - x) \frac{d}{d x} {(x^{2} + 4)}^{2}}{{(x^{2} + 4)}^{4}})$

Step 2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 4$ .

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Step 2.6.1

To apply the Chain Rule, set $u$ as $x^{2} + 4$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + 2 - x) - (2 + x) (2 - x) (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} + 4))}{{(x^{2} + 4)}^{4}})$

Step 2.6.2

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

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + 2 - x) - (2 + x) (2 - x) (2 u \frac{d}{d x} (x^{2} + 4))}{{(x^{2} + 4)}^{4}})$

Step 2.6.3

Replace all occurrences of $u$ with $x^{2} + 4$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + 2 - x) - (2 + x) (2 - x) (2 (x^{2} + 4) \frac{d}{d x} (x^{2} + 4))}{{(x^{2} + 4)}^{4}})$

Step 2.7

Simplify with factoring out.

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Step 2.7.1

Multiply $2$ by $- 1$ .

$f'' (x) = 52 (\frac{{(x^{2} + 4)}^{2} (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) ((x^{2} + 4) \frac{d}{d x} (x^{2} + 4))}{{(x^{2} + 4)}^{4}})$

Step 2.7.2

Factor $x^{2} + 4$ out of ${(x^{2} + 4)}^{2} (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) ((x^{2} + 4) \frac{d}{d x} [x^{2} + 4])$ .

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Step 2.7.2.1

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

$f'' (x) = 52 (\frac{(x^{2} + 4) ((x^{2} + 4) (- (2 + x) + 2 - x)) - 2 (2 + x) (2 - x) ((x^{2} + 4) \frac{d}{d x} (x^{2} + 4))}{{(x^{2} + 4)}^{4}})$

Step 2.7.2.2

Factor $x^{2} + 4$ out of $- 2 (2 + x) (2 - x) ((x^{2} + 4) \frac{d}{d x} [x^{2} + 4])$ .

$f'' (x) = 52 (\frac{(x^{2} + 4) ((x^{2} + 4) (- (2 + x) + 2 - x)) + (x^{2} + 4) (- 2 (2 + x) (2 - x) (\frac{d}{d x} (x^{2} + 4)))}{{(x^{2} + 4)}^{4}})$

Step 2.7.2.3

Factor $x^{2} + 4$ out of $(x^{2} + 4) ((x^{2} + 4) (- (2 + x) + 2 - x)) + (x^{2} + 4) (- 2 (2 + x) (2 - x) (\frac{d}{d x} [x^{2} + 4]))$ .

$f'' (x) = 52 (\frac{(x^{2} + 4) ((x^{2} + 4) (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) (\frac{d}{d x} (x^{2} + 4)))}{{(x^{2} + 4)}^{4}})$

Step 2.8

Cancel the common factors.

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Step 2.8.1

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

$f'' (x) = 52 (\frac{(x^{2} + 4) ((x^{2} + 4) (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) (\frac{d}{d x} (x^{2} + 4)))}{(x^{2} + 4) {(x^{2} + 4)}^{3}})$

Step 2.8.2

Cancel the common factor.

$f'' (x) = 52 (\frac{(x^{2} + 4) ((x^{2} + 4) (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) (\frac{d}{d x} (x^{2} + 4)))}{(x^{2} + 4) {(x^{2} + 4)}^{3}})$

Step 2.8.3

Rewrite the expression.

$f'' (x) = 52 (\frac{(x^{2} + 4) (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) (\frac{d}{d x} (x^{2} + 4))}{{(x^{2} + 4)}^{3}})$

Step 2.9

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

$f'' (x) = 52 (\frac{(x^{2} + 4) (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (4))}{{(x^{2} + 4)}^{3}})$

Step 2.10

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

$f'' (x) = 52 (\frac{(x^{2} + 4) (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) (2 x + \frac{d}{d x} (4))}{{(x^{2} + 4)}^{3}})$

Step 2.11

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

$f'' (x) = 52 (\frac{(x^{2} + 4) (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) (2 x + 0)}{{(x^{2} + 4)}^{3}})$

Step 2.12

Combine fractions.

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Step 2.12.1

Add $2 x$ and $0$ .

$f'' (x) = 52 (\frac{(x^{2} + 4) (- (2 + x) + 2 - x) - 2 (2 + x) (2 - x) (2 x)}{{(x^{2} + 4)}^{3}})$

Step 2.12.2

Multiply $2$ by $- 2$ .

$f'' (x) = 52 (\frac{(x^{2} + 4) (- (2 + x) + 2 - x) - 4 (2 + x) (2 - x) x}{{(x^{2} + 4)}^{3}})$

Step 2.12.3

Combine $52$ and $\frac{(x^{2} + 4) (- (2 + x) + 2 - x) - 4 (2 + x) (2 - x) x}{{(x^{2} + 4)}^{3}}$ .

$f'' (x) = \frac{52 ((x^{2} + 4) (- (2 + x) + 2 - x) - 4 (2 + x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

$f'' (x) = \frac{52 ((x^{2} + 4) (- 1 \cdot 2 - x + 2 - x) - 4 (2 + x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.2

Apply the distributive property.

$f'' (x) = \frac{52 ((x^{2} + 4) (- 1 \cdot 2 - x + 2 - x) + (- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.3

Apply the distributive property.

$f'' (x) = \frac{52 ((x^{2} + 4) (- 1 \cdot 2 - x + 2 - x)) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4

Simplify the numerator.

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Step 2.13.4.1

Simplify each term.

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Step 2.13.4.1.1

Multiply $- 1$ by $2$ .

$f'' (x) = \frac{52 ((x^{2} + 4) (- 2 - x + 2 - x)) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.2

Combine the opposite terms in $- 2 - x + 2 - x$ .

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Step 2.13.4.1.2.1

Add $- 2$ and $2$ .

$f'' (x) = \frac{52 ((x^{2} + 4) (- x + 0 - x)) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.2.2

Add $- x$ and $0$ .

$f'' (x) = \frac{52 ((x^{2} + 4) (- x - x)) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.3

Subtract $x$ from $- x$ .

$f'' (x) = \frac{52 ((x^{2} + 4) (- 2 x)) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.4

Apply the distributive property.

$f'' (x) = \frac{52 (x^{2} (- 2 x) + 4 (- 2 x)) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.5

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{52 (- 2 x^{2} x + 4 (- 2 x)) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.6

Multiply $- 2$ by $4$ .

$f'' (x) = \frac{52 (- 2 x^{2} x - 8 x) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.7

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

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Step 2.13.4.1.7.1

Move $x$ .

$f'' (x) = \frac{52 (- 2 (x \cdot x^{2}) - 8 x) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.7.2

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

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Step 2.13.4.1.7.2.1

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

$f'' (x) = \frac{52 (- 2 (x \cdot x^{2}) - 8 x) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.7.2.2

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

$f'' (x) = \frac{52 (- 2 x^{1 + 2} - 8 x) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.7.3

Add $1$ and $2$ .

$f'' (x) = \frac{52 (- 2 x^{3} - 8 x) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.8

Apply the distributive property.

$f'' (x) = \frac{52 (- 2 x^{3}) + 52 (- 8 x) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.9

Multiply $- 2$ by $52$ .

$f'' (x) = \frac{- 104 x^{3} + 52 (- 8 x) + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.10

Multiply $- 8$ by $52$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 4 \cdot 2 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.11

Multiply $- 4$ by $2$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 8 - 4 x) (2 - x) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.12

Expand $(- 8 - 4 x) (2 - x)$ using the FOIL Method.

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Step 2.13.4.1.12.1

Apply the distributive property.

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 8 (2 - x) - 4 x (2 - x)) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.12.2

Apply the distributive property.

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 8 \cdot 2 - 8 (- x) - 4 x (2 - x)) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.12.3

Apply the distributive property.

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 8 \cdot 2 - 8 (- x) - 4 x \cdot 2 - 4 x (- x)) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.13

Simplify and combine like terms.

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Step 2.13.4.1.13.1

Simplify each term.

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Step 2.13.4.1.13.1.1

Multiply $- 8$ by $2$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 16 - 8 (- x) - 4 x \cdot 2 - 4 x (- x)) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.13.1.2

Multiply $- 1$ by $- 8$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 16 + 8 x - 4 x \cdot 2 - 4 x (- x)) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.13.1.3

Multiply $2$ by $- 4$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 16 + 8 x - 8 x - 4 x (- x)) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.13.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 16 + 8 x - 8 x - 4 \cdot (- 1 x \cdot x)) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.13.1.5

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

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Step 2.13.4.1.13.1.5.1

Move $x$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 16 + 8 x - 8 x - 4 \cdot (- 1 (x \cdot x))) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.13.1.5.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 16 + 8 x - 8 x - 4 \cdot (- 1 x^{2})) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.13.1.6

Multiply $- 4$ by $- 1$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 16 + 8 x - 8 x + 4 x^{2}) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.13.2

Subtract $8 x$ from $8 x$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 16 + 0 + 4 x^{2}) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.13.3

Add $- 16$ and $0$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 ((- 16 + 4 x^{2}) x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.14

Apply the distributive property.

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 (- 16 x + 4 x^{2} x)}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.15

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

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Step 2.13.4.1.15.1

Move $x$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 (- 16 x + 4 (x \cdot x^{2}))}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.15.2

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

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Step 2.13.4.1.15.2.1

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

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 (- 16 x + 4 (x \cdot x^{2}))}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.15.2.2

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

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 (- 16 x + 4 x^{1 + 2})}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.15.3

Add $1$ and $2$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 (- 16 x + 4 x^{3})}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.16

Apply the distributive property.

$f'' (x) = \frac{- 104 x^{3} - 416 x + 52 (- 16 x) + 52 (4 x^{3})}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.17

Multiply $- 16$ by $52$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x - 832 x + 52 (4 x^{3})}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.1.18

Multiply $4$ by $52$ .

$f'' (x) = \frac{- 104 x^{3} - 416 x - 832 x + 208 x^{3}}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.2

Add $- 104 x^{3}$ and $208 x^{3}$ .

$f'' (x) = \frac{104 x^{3} - 416 x - 832 x}{{(x^{2} + 4)}^{3}}$

Step 2.13.4.3

Subtract $832 x$ from $- 416 x$ .

$f'' (x) = \frac{104 x^{3} - 1248 x}{{(x^{2} + 4)}^{3}}$

Step 2.13.5

Factor $104 x$ out of $104 x^{3} - 1248 x$ .

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Step 2.13.5.1

Factor $104 x$ out of $104 x^{3}$ .

$f'' (x) = \frac{104 x (x^{2}) - 1248 x}{{(x^{2} + 4)}^{3}}$

Step 2.13.5.2

Factor $104 x$ out of $- 1248 x$ .

$f'' (x) = \frac{104 x (x^{2}) + 104 x (- 12)}{{(x^{2} + 4)}^{3}}$

Step 2.13.5.3

Factor $104 x$ out of $104 x (x^{2}) + 104 x (- 12)$ .

$f'' (x) = \frac{104 x (x^{2} - 12)}{{(x^{2} + 4)}^{3}}$

Step 3

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

$\frac{52 (2 + x) (2 - x)}{{(x^{2} + 4)}^{2}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

Since $13$ is constant with respect to $x$ , the derivative of $\frac{13 x}{1 + 0.25 x^{2}}$ with respect to $x$ is $13 \frac{d}{d x} [\frac{x}{1 + 0.25 x^{2}}]$ .

$13 \frac{d}{d x} [\frac{x}{1 + 0.25 x^{2}}]$

Step 4.1.2

$13 \frac{(1 + 0.25 x^{2}) \frac{d}{d x} [x] - x \frac{d}{d x} [1 + 0.25 x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.3

Differentiate.

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Step 4.1.3.1

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

$13 \frac{(1 + 0.25 x^{2}) \cdot 1 - x \frac{d}{d x} [1 + 0.25 x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.3.2

Multiply $1 + 0.25 x^{2}$ by $1$ .

$13 \frac{1 + 0.25 x^{2} - x \frac{d}{d x} [1 + 0.25 x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.3.3

By the Sum Rule, the derivative of $1 + 0.25 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [0.25 x^{2}]$ .

$13 \frac{1 + 0.25 x^{2} - x (\frac{d}{d x} [1] + \frac{d}{d x} [0.25 x^{2}])}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.3.4

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

$13 \frac{1 + 0.25 x^{2} - x (0 + \frac{d}{d x} [0.25 x^{2}])}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.3.5

Add $0$ and $\frac{d}{d x} [0.25 x^{2}]$ .

$13 \frac{1 + 0.25 x^{2} - x \frac{d}{d x} [0.25 x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.3.6

Since $0.25$ is constant with respect to $x$ , the derivative of $0.25 x^{2}$ with respect to $x$ is $0.25 \frac{d}{d x} [x^{2}]$ .

$13 \frac{1 + 0.25 x^{2} - x (0.25 \frac{d}{d x} [x^{2}])}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.3.7

Multiply $0.25$ by $- 1$ .

$13 \frac{1 + 0.25 x^{2} - 0.25 x \frac{d}{d x} [x^{2}]}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.3.8

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

$13 \frac{1 + 0.25 x^{2} - 0.25 x (2 x)}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.3.9

Multiply $2$ by $- 0.25$ .

$13 \frac{1 + 0.25 x^{2} - 0.5 x \cdot x}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.4

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

$13 \frac{1 + 0.25 x^{2} - 0.5 (x^{1} x)}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.5

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

$13 \frac{1 + 0.25 x^{2} - 0.5 (x^{1} x^{1})}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.6

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

$13 \frac{1 + 0.25 x^{2} - 0.5 x^{1 + 1}}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.7

Add $1$ and $1$ .

$13 \frac{1 + 0.25 x^{2} - 0.5 x^{2}}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.8

Subtract $0.5 x^{2}$ from $0.25 x^{2}$ .

$13 \frac{1 - 0.25 x^{2}}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.9

Combine $13$ and $\frac{1 - 0.25 x^{2}}{{(1 + 0.25 x^{2})}^{2}}$ .

$\frac{13 (1 - 0.25 x^{2})}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.10

Simplify.

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Step 4.1.10.1

Apply the distributive property.

$\frac{13 \cdot 1 + 13 (- 0.25 x^{2})}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.10.2

Simplify each term.

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Step 4.1.10.2.1

Multiply $13$ by $1$ .

$\frac{13 + 13 (- 0.25 x^{2})}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.10.2.2

Multiply $- 0.25$ by $13$ .

$\frac{13 - 3.25 x^{2}}{{(1 + 0.25 x^{2})}^{2}}$

Step 4.1.10.3

Reorder terms.

$\frac{- 3.25 x^{2} + 13}{{(0.25 x^{2} + 1)}^{2}}$

Step 4.1.10.4

Simplify the numerator.

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Step 4.1.10.4.1

Factor $3.25$ out of $- 3.25 x^{2} + 13$ .

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Step 4.1.10.4.1.1

Factor $3.25$ out of $- 3.25 x^{2}$ .

$\frac{3.25 (- x^{2}) + 13}{{(0.25 x^{2} + 1)}^{2}}$

Step 4.1.10.4.1.2

Factor $3.25$ out of $13$ .

$\frac{3.25 (- x^{2}) + 3.25 (4)}{{(0.25 x^{2} + 1)}^{2}}$

Step 4.1.10.4.1.3

Factor $3.25$ out of $3.25 (- x^{2}) + 3.25 (4)$ .

$\frac{3.25 (- x^{2} + 4)}{{(0.25 x^{2} + 1)}^{2}}$

Step 4.1.10.4.2

Rewrite $4$ as $2^{2}$ .

$\frac{3.25 (- x^{2} + 2^{2})}{{(0.25 x^{2} + 1)}^{2}}$

Step 4.1.10.4.3

Reorder $- x^{2}$ and $2^{2}$ .

$\frac{3.25 (2^{2} - x^{2})}{{(0.25 x^{2} + 1)}^{2}}$

Step 4.1.10.4.4

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

$\frac{3.25 (2 + x) (2 - x)}{{(0.25 x^{2} + 1)}^{2}}$

Step 4.1.10.5

Simplify the denominator.

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Step 4.1.10.5.1

Factor $0.25$ out of $0.25 x^{2} + 1$ .

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Step 4.1.10.5.1.1

Factor $0.25$ out of $0.25 x^{2}$ .

$\frac{3.25 (2 + x) (2 - x)}{{(0.25 (x^{2}) + 1)}^{2}}$

Step 4.1.10.5.1.2

Factor $0.25$ out of $1$ .

$\frac{3.25 (2 + x) (2 - x)}{{(0.25 x^{2} + 0.25 \cdot 4)}^{2}}$

Step 4.1.10.5.1.3

Factor $0.25$ out of $0.25 x^{2} + 0.25 \cdot 4$ .

$\frac{3.25 (2 + x) (2 - x)}{{(0.25 (x^{2} + 4))}^{2}}$

Step 4.1.10.5.2

Apply the product rule to $0.25 (x^{2} + 4)$ .

$\frac{3.25 (2 + x) (2 - x)}{{0.25}^{2} {(x^{2} + 4)}^{2}}$

Step 4.1.10.5.3

Raise $0.25$ to the power of $2$ .

$\frac{3.25 (2 + x) (2 - x)}{0.0625 {(x^{2} + 4)}^{2}}$

Step 4.1.10.6

Factor $3.25$ out of $3.25 (2 + x) (2 - x)$ .

$\frac{3.25 ((2 + x) (2 - x))}{0.0625 {(x^{2} + 4)}^{2}}$

Step 4.1.10.7

Factor $0.0625$ out of $0.0625 {(x^{2} + 4)}^{2}$ .

$\frac{3.25 ((2 + x) (2 - x))}{0.0625 ({(x^{2} + 4)}^{2})}$

Step 4.1.10.8

Separate fractions.

$\frac{3.25}{0.0625} \cdot \frac{(2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$

Step 4.1.10.9

Divide $3.25$ by $0.0625$ .

$52 \frac{(2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$

Step 4.1.10.10

Combine $52$ and $\frac{(2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$ .

$f' (x) = \frac{52 (2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{52 (2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$ .

$\frac{52 (2 + x) (2 - x)}{{(x^{2} + 4)}^{2}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{52 (2 + x) (2 - x)}{{(x^{2} + 4)}^{2}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{52 (2 + x) (2 - x)}{{(x^{2} + 4)}^{2}} = 0$

Step 5.2

Set the numerator equal to zero.

$52 (2 + x) (2 - x) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

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

$2 + x = 0$

$2 - x = 0$

Step 5.3.2

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

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Step 5.3.2.1

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

$2 + x = 0$

Step 5.3.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5.3.3

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

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Step 5.3.3.1

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

$2 - x = 0$

Step 5.3.3.2

Solve $2 - x = 0$ for $x$ .

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Step 5.3.3.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 5.3.3.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 5.3.3.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 5.3.3.2.2.2

Simplify the left side.

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Step 5.3.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 5.3.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 5.3.3.2.2.3

Simplify the right side.

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Step 5.3.3.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 5.3.4

The final solution is all the values that make $52 (2 + x) (2 - x) = 0$ true.

$x = - 2, 2$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

Step 7

Critical points to evaluate.

$x = - 2, 2$

Step 8

Evaluate the second derivative at $x = - 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{104 (- 2) ({(- 2)}^{2} - 12)}{{({(- 2)}^{2} + 4)}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Multiply $104$ by $- 2$ .

$\frac{- 208 ({(- 2)}^{2} - 12)}{{({(- 2)}^{2} + 4)}^{3}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

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

$\frac{- 208 ({(- 2)}^{2} - 12)}{{(4 + 4)}^{3}}$

Step 9.2.2

Add $4$ and $4$ .

$\frac{- 208 ({(- 2)}^{2} - 12)}{8^{3}}$

Step 9.2.3

Raise $8$ to the power of $3$ .

$\frac{- 208 ({(- 2)}^{2} - 12)}{512}$

Step 9.3

Simplify the numerator.

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Step 9.3.1

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

$\frac{- 208 (4 - 12)}{512}$

Step 9.3.2

Subtract $12$ from $4$ .

$\frac{- 208 \cdot - 8}{512}$

Step 9.4

Reduce the expression by cancelling the common factors.

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Step 9.4.1

Multiply $- 208$ by $- 8$ .

$\frac{1664}{512}$

Step 9.4.2

Cancel the common factor of $1664$ and $512$ .

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Step 9.4.2.1

Factor $128$ out of $1664$ .

$\frac{128 (13)}{512}$

Step 9.4.2.2

Cancel the common factors.

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Step 9.4.2.2.1

Factor $128$ out of $512$ .

$\frac{128 \cdot 13}{128 \cdot 4}$

Step 9.4.2.2.2

Cancel the common factor.

$\frac{128 \cdot 13}{128 \cdot 4}$

Step 9.4.2.2.3

Rewrite the expression.

$\frac{13}{4}$

Step 10

$x = - 2$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - 2$ is a local minimum

Step 11

Find the y-value when $x = - 2$ .

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Step 11.1

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

$f (- 2) = \frac{13 (- 2)}{1 + 0.25 {(- 2)}^{2}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Multiply $13$ by $- 2$ .

$f (- 2) = \frac{- 26}{1 + 0.25 {(- 2)}^{2}}$

Step 11.2.2

Simplify the denominator.

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Step 11.2.2.1

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

$f (- 2) = \frac{- 26}{1 + 0.25 \cdot 4}$

Step 11.2.2.2

Multiply $0.25$ by $4$ .

$f (- 2) = \frac{- 26}{1 + 1}$

Step 11.2.2.3

Add $1$ and $1$ .

$f (- 2) = \frac{- 26}{2}$

Step 11.2.3

Divide $- 26$ by $2$ .

$f (- 2) = - 13$

Step 11.2.4

The final answer is $- 13$ .

$y = - 13$

Step 12

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{104 (2) ({(2)}^{2} - 12)}{{({(2)}^{2} + 4)}^{3}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Multiply $104$ by $2$ .

$\frac{208 (2^{2} - 12)}{{(2^{2} + 4)}^{3}}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

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

$\frac{208 (2^{2} - 12)}{{(4 + 4)}^{3}}$

Step 13.2.2

Add $4$ and $4$ .

$\frac{208 (2^{2} - 12)}{8^{3}}$

Step 13.2.3

Raise $8$ to the power of $3$ .

$\frac{208 (2^{2} - 12)}{512}$

Step 13.3

Simplify the numerator.

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Step 13.3.1

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

$\frac{208 (4 - 12)}{512}$

Step 13.3.2

Subtract $12$ from $4$ .

$\frac{208 \cdot - 8}{512}$

Step 13.4

Reduce the expression by cancelling the common factors.

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Step 13.4.1

Multiply $208$ by $- 8$ .

$\frac{- 1664}{512}$

Step 13.4.2

Cancel the common factor of $- 1664$ and $512$ .

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Step 13.4.2.1

Factor $128$ out of $- 1664$ .

$\frac{128 (- 13)}{512}$

Step 13.4.2.2

Cancel the common factors.

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Step 13.4.2.2.1

Factor $128$ out of $512$ .

$\frac{128 \cdot - 13}{128 \cdot 4}$

Step 13.4.2.2.2

Cancel the common factor.

$\frac{128 \cdot - 13}{128 \cdot 4}$

Step 13.4.2.2.3

Rewrite the expression.

$\frac{- 13}{4}$

Step 13.4.3

Move the negative in front of the fraction.

$- \frac{13}{4}$

Step 14

$x = 2$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 2$ is a local maximum

Step 15

Find the y-value when $x = 2$ .

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Step 15.1

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

$f (2) = \frac{13 (2)}{1 + 0.25 {(2)}^{2}}$

Step 15.2

Simplify the result.

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Step 15.2.1

Multiply $13$ by $2$ .

$f (2) = \frac{26}{1 + 0.25 \cdot 2^{2}}$

Step 15.2.2

Simplify the denominator.

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Step 15.2.2.1

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

$f (2) = \frac{26}{1 + 0.25 \cdot 4}$

Step 15.2.2.2

Multiply $0.25$ by $4$ .

$f (2) = \frac{26}{1 + 1}$

Step 15.2.2.3

Add $1$ and $1$ .

$f (2) = \frac{26}{2}$

Step 15.2.3

Divide $26$ by $2$ .

$f (2) = 13$

Step 15.2.4

The final answer is $13$ .

$y = 13$

Step 16

These are the local extrema for $f (x) = \frac{13 x}{1 + 0.25 x^{2}}$ .

$(- 2, - 13)$ is a local minima

$(2, 13)$ is a local maxima

Step 17