Calculus Examples

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

Step 1

Write $y = \frac{1}{\sqrt{x^{2} + 2}}$ as a function.

$f (x) = \frac{1}{\sqrt{x^{2} + 2}}$

Step 2

Find the first derivative of the function.

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Step 2.1

Apply basic rules of exponents.

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Step 2.1.1

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

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

Step 2.1.2

Rewrite $\frac{1}{{(x^{2} + 2)}^{\frac{1}{2}}}$ as ${({(x^{2} + 2)}^{\frac{1}{2}})}^{- 1}$ .

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Move the negative in front of the fraction.

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

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 $f (x) = x^{- \frac{1}{2}}$ and $g (x) = x^{2} + 2$ .

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Step 2.2.1

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

$\frac{d}{d u} [u^{- \frac{1}{2}}] \frac{d}{d x} [x^{2} + 2]$

Step 2.2.2

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

$- \frac{1}{2} u^{- \frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 2]$

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $- 1$ .

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

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

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

Step 2.7.2

Combine ${(x^{2} + 2)}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

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

Step 2.7.3

Move ${(x^{2} + 2)}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Simplify terms.

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Step 2.11.1

Add $2 x$ and $0$ .

$- \frac{1}{2 {(x^{2} + 2)}^{\frac{3}{2}}} (2 x)$

Step 2.11.2

Multiply $2$ by $- 1$ .

$- 2 \frac{1}{2 {(x^{2} + 2)}^{\frac{3}{2}}} x$

Step 2.11.3

Combine $- 2$ and $\frac{1}{2 {(x^{2} + 2)}^{\frac{3}{2}}}$ .

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

Step 2.11.4

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

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

Step 2.11.5

Factor $2$ out of $- 2 x$ .

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

Step 2.12

Cancel the common factors.

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Step 2.12.1

Factor $2$ out of $2 {(x^{2} + 2)}^{\frac{3}{2}}$ .

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

Step 2.12.2

Cancel the common factor.

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

Step 2.12.3

Rewrite the expression.

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

Step 2.13

Move the negative in front of the fraction.

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

Step 3

Find the second derivative of the function.

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Step 3.1

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 $f (x) = - 1$ and $g (x) = \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}}$ .

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

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

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

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

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

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Step 3.3.1.1

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

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

Step 3.3.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.2.1

Cancel the common factor.

Step 3.3.1.2.2

Rewrite the expression.

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

Step 3.3.2

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

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

Step 3.3.3

Multiply ${(x^{2} + 2)}^{\frac{3}{2}}$ by $1$ .

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

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

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Step 3.4.1

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

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

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

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

Step 3.4.3

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Combine the numerators over the common denominator.

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

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $2$ .

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

Step 3.8.2

Subtract $2$ from $3$ .

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

Step 3.9

Combine fractions.

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Step 3.9.1

Combine $\frac{3}{2}$ and ${(x^{2} + 2)}^{\frac{1}{2}}$ .

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

Step 3.9.2

Combine $\frac{3 {(x^{2} + 2)}^{\frac{1}{2}}}{2}$ and $x$ .

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

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

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

Step 3.13

Combine fractions.

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Step 3.13.1

Add $2 x$ and $0$ .

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

Step 3.13.2

Multiply $2$ by $- 1$ .

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

Step 3.13.3

Combine $- 2$ and $\frac{3 {(x^{2} + 2)}^{\frac{1}{2}} x}{2}$ .

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

Step 3.13.4

Multiply $3$ by $- 2$ .

$f'' (x) = - \frac{{(x^{2} + 2)}^{\frac{3}{2}} + \frac{- 6 ({(x^{2} + 2)}^{\frac{1}{2}} x)}{2} \cdot x}{{(x^{2} + 2)}^{3}} + \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.13.5

Combine $\frac{- 6 ({(x^{2} + 2)}^{\frac{1}{2}} x)}{2}$ and $x$ .

$f'' (x) = - \frac{{(x^{2} + 2)}^{\frac{3}{2}} + \frac{- 6 ({(x^{2} + 2)}^{\frac{1}{2}} x) x}{2}}{{(x^{2} + 2)}^{3}} + \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.14

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

$f'' (x) = - \frac{{(x^{2} + 2)}^{\frac{3}{2}} + \frac{- 6 ({(x^{2} + 2)}^{\frac{1}{2}} (x \cdot x))}{2}}{{(x^{2} + 2)}^{3}} + \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.15

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

$f'' (x) = - \frac{{(x^{2} + 2)}^{\frac{3}{2}} + \frac{- 6 ({(x^{2} + 2)}^{\frac{1}{2}} (x \cdot x))}{2}}{{(x^{2} + 2)}^{3}} + \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.16

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

$f'' (x) = - \frac{{(x^{2} + 2)}^{\frac{3}{2}} + \frac{- 6 ({(x^{2} + 2)}^{\frac{1}{2}} x^{1 + 1})}{2}}{{(x^{2} + 2)}^{3}} + \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.17

Add $1$ and $1$ .

$f'' (x) = - \frac{{(x^{2} + 2)}^{\frac{3}{2}} + \frac{- 6 ({(x^{2} + 2)}^{\frac{1}{2}} x^{2})}{2}}{{(x^{2} + 2)}^{3}} + \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.18

Factor $2$ out of $- 6 {(x^{2} + 2)}^{\frac{1}{2}} x^{2}$ .

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

Step 3.19

Cancel the common factors.

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Step 3.19.1

Factor $2$ out of $2$ .

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

Step 3.19.2

Cancel the common factor.

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

Step 3.19.3

Rewrite the expression.

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

Step 3.19.4

Divide $- 3 {(x^{2} + 2)}^{\frac{1}{2}} x^{2}$ by $1$ .

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

Step 3.20

Factor ${(x^{2} + 2)}^{\frac{1}{2}}$ out of ${(x^{2} + 2)}^{\frac{3}{2}} - 3 {(x^{2} + 2)}^{\frac{1}{2}} x^{2}$ .

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Step 3.20.1

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

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

Step 3.20.2

Factor ${(x^{2} + 2)}^{\frac{1}{2}}$ out of ${(x^{2} + 2)}^{\frac{3}{2}}$ .

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

Step 3.20.3

Factor ${(x^{2} + 2)}^{\frac{1}{2}}$ out of $- 3 x^{2} {(x^{2} + 2)}^{\frac{1}{2}}$ .

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

Step 3.20.4

Factor ${(x^{2} + 2)}^{\frac{1}{2}}$ out of ${(x^{2} + 2)}^{\frac{1}{2}} {(x^{2} + 2)}^{\frac{2}{2}} + {(x^{2} + 2)}^{\frac{1}{2}} (- 3 x^{2})$ .

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

Step 3.21

Cancel the common factor of $2$ .

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Step 3.21.1

Cancel the common factor.

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

Step 3.21.2

Rewrite the expression.

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

Step 3.22

Simplify.

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

Step 3.23

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

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

Step 3.24

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

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

Step 3.25

Multiply ${(x^{2} + 2)}^{3}$ by ${(x^{2} + 2)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.25.1

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

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

Step 3.25.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.25.3

Combine $3$ and $\frac{2}{2}$ .

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

Step 3.25.4

Combine the numerators over the common denominator.

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

Step 3.25.5

Simplify the numerator.

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Step 3.25.5.1

Multiply $3$ by $2$ .

$f'' (x) = - \frac{- 2 x^{2} + 2}{{(x^{2} + 2)}^{\frac{6 - 1}{2}}} + \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.25.5.2

Subtract $1$ from $6$ .

$f'' (x) = - \frac{- 2 x^{2} + 2}{{(x^{2} + 2)}^{\frac{5}{2}}} + \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}} \cdot \frac{d}{d x} (- 1)$

Step 3.26

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

$f'' (x) = - \frac{- 2 x^{2} + 2}{{(x^{2} + 2)}^{\frac{5}{2}}} + \frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}} \cdot 0$

Step 3.27

Simplify the expression.

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Step 3.27.1

Multiply $\frac{x}{{(x^{2} + 2)}^{\frac{3}{2}}}$ by $0$ .

$f'' (x) = - \frac{- 2 x^{2} + 2}{{(x^{2} + 2)}^{\frac{5}{2}}} + 0$

Step 3.27.2

Add $- \frac{- 2 x^{2} + 2}{{(x^{2} + 2)}^{\frac{5}{2}}}$ and $0$ .

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

Step 4

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

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

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

Apply basic rules of exponents.

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Step 5.1.1.1

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

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

Step 5.1.1.2

Rewrite $\frac{1}{{(x^{2} + 2)}^{\frac{1}{2}}}$ as ${({(x^{2} + 2)}^{\frac{1}{2}})}^{- 1}$ .

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

Step 5.1.1.3

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

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Step 5.1.1.3.1

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

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

Step 5.1.1.3.2

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

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

Step 5.1.1.3.3

Move the negative in front of the fraction.

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

Step 5.1.2

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

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Step 5.1.2.1

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

$\frac{d}{d u} [u^{- \frac{1}{2}}] \frac{d}{d x} [x^{2} + 2]$

Step 5.1.2.2

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

$- \frac{1}{2} u^{- \frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 2]$

Step 5.1.2.3

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

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

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

Combine the numerators over the common denominator.

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

Step 5.1.6

Simplify the numerator.

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Step 5.1.6.1

Multiply $- 1$ by $2$ .

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

Step 5.1.6.2

Subtract $2$ from $- 1$ .

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

Step 5.1.7

Combine fractions.

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Step 5.1.7.1

Move the negative in front of the fraction.

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

Step 5.1.7.2

Combine ${(x^{2} + 2)}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

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

Step 5.1.7.3

Move ${(x^{2} + 2)}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.1.8

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

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

Step 5.1.9

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

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

Step 5.1.10

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

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

Step 5.1.11

Simplify terms.

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Step 5.1.11.1

Add $2 x$ and $0$ .

$- \frac{1}{2 {(x^{2} + 2)}^{\frac{3}{2}}} (2 x)$

Step 5.1.11.2

Multiply $2$ by $- 1$ .

$- 2 \frac{1}{2 {(x^{2} + 2)}^{\frac{3}{2}}} x$

Step 5.1.11.3

Combine $- 2$ and $\frac{1}{2 {(x^{2} + 2)}^{\frac{3}{2}}}$ .

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

Step 5.1.11.4

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

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

Step 5.1.11.5

Factor $2$ out of $- 2 x$ .

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

Step 5.1.12

Cancel the common factors.

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Step 5.1.12.1

Factor $2$ out of $2 {(x^{2} + 2)}^{\frac{3}{2}}$ .

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

Step 5.1.12.2

Cancel the common factor.

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

Step 5.1.12.3

Rewrite the expression.

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

Step 5.1.13

Move the negative in front of the fraction.

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

Step 5.2

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

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

Step 6

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

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Step 6.1

Set the first derivative equal to $0$ .

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

Step 6.2

Set the numerator equal to zero.

$x = 0$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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 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.

$- \frac{- 2 {(0)}^{2} + 2}{{({(0)}^{2} + 2)}^{\frac{5}{2}}}$

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 $0$ .

$- \frac{- 2 \cdot 0 + 2}{{(0^{2} + 2)}^{\frac{5}{2}}}$

Step 10.1.2

Multiply $- 2$ by $0$ .

$- \frac{0 + 2}{{(0^{2} + 2)}^{\frac{5}{2}}}$

Step 10.1.3

Add $0$ and $2$ .

$- \frac{2}{{(0^{2} + 2)}^{\frac{5}{2}}}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

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

$- \frac{2}{{(0 + 2)}^{\frac{5}{2}}}$

Step 10.2.2

Add $0$ and $2$ .

$- \frac{2}{2^{\frac{5}{2}}}$

Step 10.3

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

$- \frac{1}{2^{\frac{5}{2}} \cdot 2^{- 1}}$

Step 10.4

Multiply $2^{\frac{5}{2}}$ by $2^{- 1}$ by adding the exponents.

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Step 10.4.1

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

$- \frac{1}{2^{\frac{5}{2} - 1}}$

Step 10.4.2

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

$- \frac{1}{2^{\frac{5}{2} - 1 \cdot \frac{2}{2}}}$

Step 10.4.3

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

$- \frac{1}{2^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}}}$

Step 10.4.4

Combine the numerators over the common denominator.

$- \frac{1}{2^{\frac{5 - 1 \cdot 2}{2}}}$

Step 10.4.5

Simplify the numerator.

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Step 10.4.5.1

Multiply $- 1$ by $2$ .

$- \frac{1}{2^{\frac{5 - 2}{2}}}$

Step 10.4.5.2

Subtract $2$ from $5$ .

$- \frac{1}{2^{\frac{3}{2}}}$

Step 11

$x = 0$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0$ is a local maximum

Step 12

Find the y-value when $x = 0$ .

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Step 12.1

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

$f (0) = \frac{1}{\sqrt{{(0)}^{2} + 2}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify the denominator.

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Step 12.2.1.1

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

$f (0) = \frac{1}{\sqrt{0 + 2}}$

Step 12.2.1.2

Add $0$ and $2$ .

$f (0) = \frac{1}{\sqrt{2}}$

Step 12.2.2

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (0) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 12.2.3

Combine and simplify the denominator.

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Step 12.2.3.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$f (0) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 12.2.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$f (0) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 12.2.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$f (0) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 12.2.3.4

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

$f (0) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 12.2.3.5

Add $1$ and $1$ .

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

Step 12.2.3.6

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

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Step 12.2.3.6.1

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

$f (0) = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 12.2.3.6.2

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

$f (0) = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 12.2.3.6.3

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

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

Step 12.2.3.6.4

Cancel the common factor of $2$ .

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Step 12.2.3.6.4.1

Cancel the common factor.

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

Step 12.2.3.6.4.2

Rewrite the expression.

$f (0) = \frac{\sqrt{2}}{2}$

Step 12.2.3.6.5

Evaluate the exponent.

$f (0) = \frac{\sqrt{2}}{2}$

Step 12.2.4

The final answer is $\frac{\sqrt{2}}{2}$ .

$y = \frac{\sqrt{2}}{2}$

Step 13

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

$(0, \frac{\sqrt{2}}{2})$ is a local maxima

Step 14