Calculus Examples

$\sqrt{x^{2} + 4 x + 8}$

Step 1

Write $\sqrt{x^{2} + 4 x + 8}$ as a function.

$f (x) = \sqrt{x^{2} + 4 x + 8}$

Step 2

Find the first derivative of the function.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 4 x + 8}$ as ${(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{2} + 4 x + 8)}^{\frac{1}{2}}]$

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 $g (x) = x^{2} + 4 x + 8$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 4 x + 8$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.2.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.2.3

Replace all occurrences of $u$ with $x^{2} + 4 x + 8$ .

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.3

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

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.4

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

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.7

Combine fractions.

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Step 2.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.7.2

Combine $\frac{1}{2}$ and ${(x^{2} + 4 x + 8)}^{- \frac{1}{2}}$ .

$\frac{{(x^{2} + 4 x + 8)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.7.3

Move ${(x^{2} + 4 x + 8)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 2.8

By the Sum Rule, the derivative of $x^{2} + 4 x + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [8]$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [8])$

Step 2.9

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

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [8])$

Step 2.10

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [8])$

Step 2.11

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

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4 \cdot 1 + \frac{d}{d x} [8])$

Step 2.12

Multiply $4$ by $1$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4 + \frac{d}{d x} [8])$

Step 2.13

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4 + 0)$

Step 2.14

Add $2 x + 4$ and $0$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4)$

Step 2.15

Simplify.

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Step 2.15.1

Reorder the factors of $\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4)$ .

$(2 x + 4) \frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 2.15.2

Multiply $2 x + 4$ by $\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$ .

$\frac{2 x + 4}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 2.15.3

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 4}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 2.15.4

Factor $2$ out of $4$ .

$\frac{2 (x) + 2 \cdot 2}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 2.15.5

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

$\frac{2 (x + 2)}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 2.15.6

Cancel the common factors.

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Step 2.15.6.1

Factor $2$ out of $2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

$\frac{2 (x + 2)}{2 ({(x^{2} + 4 x + 8)}^{\frac{1}{2}})}$

Step 2.15.6.2

Cancel the common factor.

$\frac{2 (x + 2)}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 2.15.6.3

Rewrite the expression.

$\frac{x + 2}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 3

Find the second derivative of the function.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x + 2$ and $g (x) = {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

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

Step 3.2

Multiply the exponents in ${({(x^{2} + 4 x + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Rewrite the expression.

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

Step 3.3

Simplify.

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

$f'' (x) = \frac{{(x^{2} + 4 x + 8)}^{\frac{1}{2}} (1 + 0) - (x + 2) \frac{d}{d x} {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}{x^{2} + 4 x + 8}$

Step 3.4.4

Simplify the expression.

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Step 3.4.4.1

Add $1$ and $0$ .

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

Step 3.4.4.2

Multiply ${(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ by $1$ .

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

Step 3.5

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 $g (x) = x^{2} + 4 x + 8$ .

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Step 3.5.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + 4 x + 8$ .

$f'' (x) = \frac{{(x^{2} + 4 x + 8)}^{\frac{1}{2}} - (x + 2) (\frac{d}{d u (1)} ({u_{1}}^{\frac{1}{2}}) \frac{d}{d x} (x^{2} + 4 x + 8))}{x^{2} + 4 x + 8}$

Step 3.5.2

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

$f'' (x) = \frac{{(x^{2} + 4 x + 8)}^{\frac{1}{2}} - (x + 2) (\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} (x^{2} + 4 x + 8))}{x^{2} + 4 x + 8}$

Step 3.5.3

Replace all occurrences of $u_{1}$ with $x^{2} + 4 x + 8$ .

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Combine the numerators over the common denominator.

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

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $- 1$ by $2$ .

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

Step 3.9.2

Subtract $2$ from $1$ .

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

Step 3.10

Combine fractions.

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Step 3.10.1

Move the negative in front of the fraction.

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

Step 3.10.2

Combine $\frac{1}{2}$ and ${(x^{2} + 4 x + 8)}^{- \frac{1}{2}}$ .

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

Step 3.10.3

Move ${(x^{2} + 4 x + 8)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.11

By the Sum Rule, the derivative of $x^{2} + 4 x + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [8]$ .

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

Step 3.12

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

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

Step 3.13

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 3.14

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

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

Step 3.15

Multiply $4$ by $1$ .

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

Step 3.16

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f'' (x) = \frac{{(x^{2} + 4 x + 8)}^{\frac{1}{2}} - (x + 2) (\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} \cdot (2 x + 4 + 0))}{x^{2} + 4 x + 8}$

Step 3.17

Add $2 x + 4$ and $0$ .

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

Step 3.18

Simplify.

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Step 3.18.1

Apply the distributive property.

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

Step 3.18.2

Simplify the numerator.

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Step 3.18.2.1

Let $u_{2} = {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ . Substitute $u_{2}$ for all occurrences of ${(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

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Step 3.18.2.1.1

Raise $u_{2}$ to the power of $1$ .

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

Step 3.18.2.1.2

Raise $u_{2}$ to the power of $1$ .

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

Step 3.18.2.1.3

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

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

Step 3.18.2.1.4

Add $1$ and $1$ .

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

Step 3.18.2.1.5

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

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

Step 3.18.2.1.6

Simplify.

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Step 3.18.2.1.6.1

Apply the distributive property.

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

Step 3.18.2.1.6.2

Multiply $- 1$ by $2$ .

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

Step 3.18.2.2

Replace all occurrences of $u_{2}$ with ${(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

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

Step 3.18.2.3

Simplify.

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Step 3.18.2.3.1

Expand $({(x^{2} + 4 x + 8)}^{\frac{1}{2}} + x + 2) ({(x^{2} + 4 x + 8)}^{\frac{1}{2}} - x - 2)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.18.2.3.2

Combine the opposite terms in ${(x^{2} + 4 x + 8)}^{\frac{1}{2}} {(x^{2} + 4 x + 8)}^{\frac{1}{2}} + {(x^{2} + 4 x + 8)}^{\frac{1}{2}} (- x) + {(x^{2} + 4 x + 8)}^{\frac{1}{2}} \cdot - 2 + x {(x^{2} + 4 x + 8)}^{\frac{1}{2}} + x (- x) + x \cdot - 2 + 2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}} + 2 (- x) + 2 \cdot - 2$ .

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Step 3.18.2.3.2.1

Reorder the factors in the terms ${(x^{2} + 4 x + 8)}^{\frac{1}{2}} (- x)$ and $x {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

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

Step 3.18.2.3.2.2

Add $- x {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ and $x {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{{(x^{2} + 4 x + 8)}^{\frac{1}{2}} {(x^{2} + 4 x + 8)}^{\frac{1}{2}} + {(x^{2} + 4 x + 8)}^{\frac{1}{2}} \cdot - 2 + 0 + x (- x) + x \cdot - 2 + 2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}} + 2 (- x) + 2 \cdot - 2}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}}}{x^{2} + 4 x + 8}$

Step 3.18.2.3.2.3

Add ${(x^{2} + 4 x + 8)}^{\frac{1}{2}} {(x^{2} + 4 x + 8)}^{\frac{1}{2}} + {(x^{2} + 4 x + 8)}^{\frac{1}{2}} \cdot - 2$ and $0$ .

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

Step 3.18.2.3.2.4

Reorder the factors in the terms ${(x^{2} + 4 x + 8)}^{\frac{1}{2}} \cdot - 2$ and $2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

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

Step 3.18.2.3.2.5

Add $- 2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ and $2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

$f'' (x) = \frac{\frac{{(x^{2} + 4 x + 8)}^{\frac{1}{2}} {(x^{2} + 4 x + 8)}^{\frac{1}{2}} + x (- x) + x \cdot - 2 + 0 + 2 (- x) + 2 \cdot - 2}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}}}{x^{2} + 4 x + 8}$

Step 3.18.2.3.2.6

Add ${(x^{2} + 4 x + 8)}^{\frac{1}{2}} {(x^{2} + 4 x + 8)}^{\frac{1}{2}} + x (- x) + x \cdot - 2$ and $0$ .

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

Step 3.18.2.3.3

Simplify each term.

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Step 3.18.2.3.3.1

Multiply ${(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ by ${(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ by adding the exponents.

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Step 3.18.2.3.3.1.1

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

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

Step 3.18.2.3.3.1.2

Combine the numerators over the common denominator.

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

Step 3.18.2.3.3.1.3

Add $1$ and $1$ .

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

Step 3.18.2.3.3.1.4

Divide $2$ by $2$ .

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

Step 3.18.2.3.3.2

Simplify ${(x^{2} + 4 x + 8)}^{1}$ .

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

Step 3.18.2.3.3.3

Rewrite using the commutative property of multiplication.

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

Step 3.18.2.3.3.4

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

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Step 3.18.2.3.3.4.1

Move $x$ .

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

Step 3.18.2.3.3.4.2

Multiply $x$ by $x$ .

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

Step 3.18.2.3.3.5

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

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

Step 3.18.2.3.3.6

Multiply $- 1$ by $2$ .

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

Step 3.18.2.3.3.7

Multiply $2$ by $- 2$ .

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

Step 3.18.2.3.4

Combine the opposite terms in $x^{2} + 4 x + 8 - x^{2} - 2 x - 2 x - 4$ .

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Step 3.18.2.3.4.1

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

$f'' (x) = \frac{\frac{4 x + 8 + 0 - 2 x - 2 x - 4}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}}}{x^{2} + 4 x + 8}$

Step 3.18.2.3.4.2

Add $4 x + 8$ and $0$ .

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

Step 3.18.2.3.5

Subtract $2 x$ from $4 x$ .

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

Step 3.18.2.3.6

Combine the opposite terms in $2 x + 8 - 2 x - 4$ .

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Step 3.18.2.3.6.1

Subtract $2 x$ from $2 x$ .

$f'' (x) = \frac{\frac{0 + 8 - 4}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}}}{x^{2} + 4 x + 8}$

Step 3.18.2.3.6.2

Add $0$ and $8$ .

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

Step 3.18.2.3.7

Subtract $4$ from $8$ .

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

Step 3.18.3

Combine terms.

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Step 3.18.3.1

Rewrite $\frac{\frac{4}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}}}{x^{2} + 4 x + 8}$ as a product.

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

Step 3.18.3.2

Multiply $\frac{4}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$ by $\frac{1}{x^{2} + 4 x + 8}$ .

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

Step 3.18.3.3

Multiply ${(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ by $x^{2} + 4 x + 8$ by adding the exponents.

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Step 3.18.3.3.1

Multiply ${(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ by $x^{2} + 4 x + 8$ .

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Step 3.18.3.3.1.1

Raise $x^{2} + 4 x + 8$ to the power of $1$ .

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

Step 3.18.3.3.1.2

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

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

Step 3.18.3.3.2

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

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

Step 3.18.3.3.3

Combine the numerators over the common denominator.

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

Step 3.18.3.3.4

Add $1$ and $2$ .

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

Step 4

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

$\frac{x + 2}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}} = 0$

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 4 x + 8}$ as ${(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [{(x^{2} + 4 x + 8)}^{\frac{1}{2}}]$

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 $g (x) = x^{2} + 4 x + 8$ .

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Step 5.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 4 x + 8$ .

$\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} + 4 x + 8]$

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 $n = \frac{1}{2}$ .

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.2.3

Replace all occurrences of $u$ with $x^{2} + 4 x + 8$ .

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.3

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

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.4

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

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.6

Simplify the numerator.

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Step 5.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.7

Combine fractions.

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Step 5.1.7.1

Move the negative in front of the fraction.

$\frac{1}{2} {(x^{2} + 4 x + 8)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.7.2

Combine $\frac{1}{2}$ and ${(x^{2} + 4 x + 8)}^{- \frac{1}{2}}$ .

$\frac{{(x^{2} + 4 x + 8)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.7.3

Move ${(x^{2} + 4 x + 8)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} + 4 x + 8]$

Step 5.1.8

By the Sum Rule, the derivative of $x^{2} + 4 x + 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [8]$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [8])$

Step 5.1.9

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

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [8])$

Step 5.1.10

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [8])$

Step 5.1.11

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

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4 \cdot 1 + \frac{d}{d x} [8])$

Step 5.1.12

Multiply $4$ by $1$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4 + \frac{d}{d x} [8])$

Step 5.1.13

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4 + 0)$

Step 5.1.14

Add $2 x + 4$ and $0$ .

$\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4)$

Step 5.1.15

Simplify.

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Step 5.1.15.1

Reorder the factors of $\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}} (2 x + 4)$ .

$(2 x + 4) \frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 5.1.15.2

Multiply $2 x + 4$ by $\frac{1}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$ .

$\frac{2 x + 4}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 5.1.15.3

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 4}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 5.1.15.4

Factor $2$ out of $4$ .

$\frac{2 (x) + 2 \cdot 2}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 5.1.15.5

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

$\frac{2 (x + 2)}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 5.1.15.6

Cancel the common factors.

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Step 5.1.15.6.1

Factor $2$ out of $2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}$ .

$\frac{2 (x + 2)}{2 ({(x^{2} + 4 x + 8)}^{\frac{1}{2}})}$

Step 5.1.15.6.2

Cancel the common factor.

$\frac{2 (x + 2)}{2 {(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 5.1.15.6.3

Rewrite the expression.

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

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x + 2}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$ .

$\frac{x + 2}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}}$

Step 6

Set the first derivative equal to $0$ then solve the equation $\frac{x + 2}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}} = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$\frac{x + 2}{{(x^{2} + 4 x + 8)}^{\frac{1}{2}}} = 0$

Step 6.2

Set the numerator equal to zero.

$x + 2 = 0$

Step 6.3

Subtract $2$ from both sides of the equation.

$x = - 2$

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.

$x = - 2$

Step 9

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.

$\frac{4}{{({(- 2)}^{2} + 4 (- 2) + 8)}^{\frac{3}{2}}}$

Step 10

Evaluate the second derivative.

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Step 10.1

Simplify the denominator.

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Step 10.1.1

Simplify each term.

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Step 10.1.1.1

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

$\frac{4}{{(4 + 4 (- 2) + 8)}^{\frac{3}{2}}}$

Step 10.1.1.2

Multiply $4$ by $- 2$ .

$\frac{4}{{(4 - 8 + 8)}^{\frac{3}{2}}}$

Step 10.1.2

Subtract $8$ from $4$ .

$\frac{4}{{(- 4 + 8)}^{\frac{3}{2}}}$

Step 10.1.3

Add $- 4$ and $8$ .

$\frac{4}{4^{\frac{3}{2}}}$

Step 10.1.4

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

$\frac{4}{{(2^{2})}^{\frac{3}{2}}}$

Step 10.1.5

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

$\frac{4}{2^{2 (\frac{3}{2})}}$

Step 10.1.6

Cancel the common factor of $2$ .

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Step 10.1.6.1

Cancel the common factor.

$\frac{4}{2^{2 (\frac{3}{2})}}$

Step 10.1.6.2

Rewrite the expression.

$\frac{4}{2^{3}}$

Step 10.1.7

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

$\frac{4}{8}$

Step 10.2

Cancel the common factor of $4$ and $8$ .

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Step 10.2.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{8}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 1}{4 \cdot 2}$

Step 10.2.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 2}$

Step 10.2.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 11

$x = - 2$ 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 12

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

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Step 12.1

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

$f (- 2) = \sqrt{{(- 2)}^{2} + 4 (- 2) + 8}$

Step 12.2

Simplify the result.

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Step 12.2.1

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

$f (- 2) = \sqrt{4 + 4 (- 2) + 8}$

Step 12.2.2

Multiply $4$ by $- 2$ .

$f (- 2) = \sqrt{4 - 8 + 8}$

Step 12.2.3

Subtract $8$ from $4$ .

$f (- 2) = \sqrt{- 4 + 8}$

Step 12.2.4

Add $- 4$ and $8$ .

$f (- 2) = \sqrt{4}$

Step 12.2.5

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

$f (- 2) = \sqrt{2^{2}}$

Step 12.2.6

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

$f (- 2) = 2$

Step 12.2.7

The final answer is $2$ .

$y = 2$

Step 13

These are the local extrema for $f (x) = \sqrt{x^{2} + 4 x + 8}$ .

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

Step 14