Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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 $f (x) = x$ and $g (x) = {(8 - x^{2})}^{\frac{1}{2}}$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

Combine the numerators over the common denominator.

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

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

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

Step 1.1.7.2

Subtract $2$ from $1$ .

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

Step 1.1.8

Combine fractions.

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Step 1.1.8.1

Move the negative in front of the fraction.

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

Step 1.1.8.2

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

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

Step 1.1.8.3

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

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

Step 1.1.8.4

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

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

Step 1.1.9

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

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

Step 1.1.10

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

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

Step 1.1.11

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

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

Step 1.1.12

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

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

Step 1.1.13

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

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

Step 1.1.14

Combine fractions.

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Step 1.1.14.1

Multiply $2$ by $- 1$ .

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

Step 1.1.14.2

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

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

Step 1.1.14.3

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

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

Step 1.1.15

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

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

Step 1.1.16

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

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

Step 1.1.17

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

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

Step 1.1.18

Add $1$ and $1$ .

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

Step 1.1.19

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

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

Step 1.1.20

Cancel the common factors.

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Step 1.1.20.1

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

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

Step 1.1.20.2

Cancel the common factor.

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

Step 1.1.20.3

Rewrite the expression.

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

Step 1.1.21

Move the negative in front of the fraction.

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

Step 1.1.22

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

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

Step 1.1.23

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

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

Step 1.1.24

To write ${(8 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(8 - x^{2})}^{\frac{1}{2}}}{{(8 - x^{2})}^{\frac{1}{2}}}$ .

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

Step 1.1.25

Combine the numerators over the common denominator.

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

Step 1.1.26

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

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Step 1.1.26.1

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

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

Step 1.1.26.2

Combine the numerators over the common denominator.

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

Step 1.1.26.3

Add $1$ and $1$ .

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

Step 1.1.26.4

Divide $2$ by $2$ .

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

Step 1.1.27

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

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

Step 1.1.28

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

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

Step 1.1.29

Reorder terms.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$- 2 x^{2} + 8 = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Subtract $8$ from both sides of the equation.

$- 2 x^{2} = - 8$

Step 2.3.2

Divide each term in $- 2 x^{2} = - 8$ by $- 2$ and simplify.

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Step 2.3.2.1

Divide each term in $- 2 x^{2} = - 8$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} = \frac{- 8}{- 2}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} = \frac{- 8}{- 2}$

Step 2.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 8}{- 2}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $- 8$ by $- 2$ .

$x^{2} = 4$

Step 2.3.3

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

$x = \pm \sqrt{4}$

Step 2.3.4

Simplify $\pm \sqrt{4}$ .

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Step 2.3.4.1

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

$x = \pm \sqrt{2^{2}}$

Step 2.3.4.2

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

$x = \pm 2$

Step 2.3.5

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

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Step 2.3.5.1

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

$x = 2$

Step 2.3.5.2

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

$x = - 2$

Step 2.3.5.3

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

$x = 2, - 2$

Step 3

The values which make the derivative equal to $0$ are $2, - 2$ .

$2, - 2$

Step 4

Find where the derivative is undefined.

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Step 4.1

Convert expressions with fractional exponents to radicals.

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Step 4.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 4.1.2

Anything raised to $1$ is the base itself.

$\frac{- 2 x^{2} + 8}{\sqrt{- x^{2} + 8}}$

Step 4.2

Set the denominator in $\frac{- 2 x^{2} + 8}{\sqrt{- x^{2} + 8}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{- x^{2} + 8} = 0$

Step 4.3

Solve for $x$ .

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Step 4.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{- x^{2} + 8}}^{2} = 0^{2}$

Step 4.3.2

Simplify each side of the equation.

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Step 4.3.2.1

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

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

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Simplify ${({(- x^{2} + 8)}^{\frac{1}{2}})}^{2}$ .

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Step 4.3.2.2.1.1

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

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Step 4.3.2.2.1.1.1

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

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

Step 4.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.1.2.1

Cancel the common factor.

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

Step 4.3.2.2.1.1.2.2

Rewrite the expression.

${(- x^{2} + 8)}^{1} = 0^{2}$

Step 4.3.2.2.1.2

Simplify.

$- x^{2} + 8 = 0^{2}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

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

$- x^{2} + 8 = 0$

Step 4.3.3

Solve for $x$ .

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Step 4.3.3.1

Subtract $8$ from both sides of the equation.

$- x^{2} = - 8$

Step 4.3.3.2

Divide each term in $- x^{2} = - 8$ by $- 1$ and simplify.

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Step 4.3.3.2.1

Divide each term in $- x^{2} = - 8$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 8}{- 1}$

Step 4.3.3.2.2

Simplify the left side.

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Step 4.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 8}{- 1}$

Step 4.3.3.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 8}{- 1}$

Step 4.3.3.2.3

Simplify the right side.

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Step 4.3.3.2.3.1

Divide $- 8$ by $- 1$ .

$x^{2} = 8$

Step 4.3.3.3

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

$x = \pm \sqrt{8}$

Step 4.3.3.4

Simplify $\pm \sqrt{8}$ .

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Step 4.3.3.4.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.3.3.4.1.1

Factor $4$ out of $8$ .

$x = \pm \sqrt{4 (2)}$

Step 4.3.3.4.1.2

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

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

Step 4.3.3.4.2

Pull terms out from under the radical.

$x = \pm 2 \sqrt{2}$

Step 4.3.3.5

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

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Step 4.3.3.5.1

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

$x = 2 \sqrt{2}$

Step 4.3.3.5.2

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

$x = - 2 \sqrt{2}$

Step 4.3.3.5.3

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

$x = 2 \sqrt{2}, - 2 \sqrt{2}$

Step 4.4

Set the radicand in $\sqrt{- x^{2} + 8}$ less than $0$ to find where the expression is undefined.

$- x^{2} + 8 < 0$

Step 4.5

Solve for $x$ .

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Step 4.5.1

Subtract $8$ from both sides of the inequality.

$- x^{2} < - 8$

Step 4.5.2

Divide each term in $- x^{2} < - 8$ by $- 1$ and simplify.

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Step 4.5.2.1

Divide each term in $- x^{2} < - 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} > \frac{- 8}{- 1}$

Step 4.5.2.2

Simplify the left side.

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Step 4.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} > \frac{- 8}{- 1}$

Step 4.5.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} > \frac{- 8}{- 1}$

Step 4.5.2.3

Simplify the right side.

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Step 4.5.2.3.1

Divide $- 8$ by $- 1$ .

$x^{2} > 8$

Step 4.5.3

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

$\sqrt{x^{2}} > \sqrt{8}$

Step 4.5.4

Simplify the equation.

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Step 4.5.4.1

Simplify the left side.

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Step 4.5.4.1.1

Pull terms out from under the radical.

$| x | > \sqrt{8}$

Step 4.5.4.2

Simplify the right side.

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Step 4.5.4.2.1

Simplify $\sqrt{8}$ .

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Step 4.5.4.2.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.5.4.2.1.1.1

Factor $4$ out of $8$ .

$| x | > \sqrt{4 (2)}$

Step 4.5.4.2.1.1.2

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

$| x | > \sqrt{2^{2} \cdot 2}$

Step 4.5.4.2.1.2

Pull terms out from under the radical.

$| x | > | 2 | \sqrt{2}$

Step 4.5.4.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | > 2 \sqrt{2}$

Step 4.5.5

Write $| x | > 2 \sqrt{2}$ as a piecewise.

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Step 4.5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 4.5.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x > 2 \sqrt{2}$

Step 4.5.5.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 4.5.5.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 2 \sqrt{2}$

Step 4.5.5.5

Write as a piecewise.

${\begin{cases} x > 2 \sqrt{2} & x \geq 0 \\ - x > 2 \sqrt{2} & x < 0 \end{cases}$

Step 4.5.6

Find the intersection of $x > 2 \sqrt{2}$ and $x \geq 0$ .

$x > 2 \sqrt{2}$

Step 4.5.7

Divide each term in $- x > 2 \sqrt{2}$ by $- 1$ and simplify.

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Step 4.5.7.1

Divide each term in $- x > 2 \sqrt{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{2 \sqrt{2}}{- 1}$

Step 4.5.7.2

Simplify the left side.

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Step 4.5.7.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{2 \sqrt{2}}{- 1}$

Step 4.5.7.2.2

Divide $x$ by $1$ .

$x < \frac{2 \sqrt{2}}{- 1}$

Step 4.5.7.3

Simplify the right side.

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Step 4.5.7.3.1

Move the negative one from the denominator of $\frac{2 \sqrt{2}}{- 1}$ .

$x < - 1 \cdot (2 \sqrt{2})$

Step 4.5.7.3.2

Rewrite $- 1 \cdot (2 \sqrt{2})$ as $- (2 \sqrt{2})$ .

$x < - (2 \sqrt{2})$

Step 4.5.7.3.3

Multiply $2$ by $- 1$ .

$x < - 2 \sqrt{2}$

Step 4.5.8

Find the union of the solutions.

$x < - 2 \sqrt{2}$ or $x > 2 \sqrt{2}$

Step 4.6

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

$x \leq - 2 \sqrt{2}, x \geq 2 \sqrt{2}$

$(- \infty, - 2 \sqrt{2}] \cup [2 \sqrt{2}, \infty)$

$x \leq - 2 \sqrt{2}, x \geq 2 \sqrt{2}$

$(- \infty, - 2 \sqrt{2}] \cup [2 \sqrt{2}, \infty)$

Step 5

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

$(- \infty, - 2 \sqrt{2}) \cup (- 2 \sqrt{2}, - 2) \cup (- 2, 2) \cup (2, 2 \sqrt{2}) \cup (2 \sqrt{2}, \infty)$

Step 6

Substitute a value from the interval $(- \infty, - 2 \sqrt{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 3.828427$ in the expression.

$f' (- 3.828427) = \frac{- 2 {(- 3.828427)}^{2} + 8}{{(- {(- 3.828427)}^{2} + 8)}^{\frac{1}{2}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Raise $- 3.828427$ to the power of $2$ .

$f' (- 3.828427) = \frac{- 2 \cdot 14.65685329 + 8}{{(- {(- 3.828427)}^{2} + 8)}^{\frac{1}{2}}}$

Step 6.2.1.2

Multiply $- 2$ by $14.65685329$ .

$f' (- 3.828427) = \frac{- 29.31370658 + 8}{{(- {(- 3.828427)}^{2} + 8)}^{\frac{1}{2}}}$

Step 6.2.1.3

Add $- 29.31370658$ and $8$ .

$f' (- 3.828427) = \frac{- 21.31370658}{{(- {(- 3.828427)}^{2} + 8)}^{\frac{1}{2}}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Raise $- 3.828427$ to the power of $2$ .

$f' (- 3.828427) = \frac{- 21.31370658}{{(- 1 \cdot 14.65685329 + 8)}^{\frac{1}{2}}}$

Step 6.2.2.1.2

Multiply $- 1$ by $14.65685329$ .

$f' (- 3.828427) = \frac{- 21.31370658}{{(- 14.65685329 + 8)}^{\frac{1}{2}}}$

Step 6.2.2.2

Add $- 14.65685329$ and $8$ .

$f' (- 3.828427) = \frac{- 21.31370658}{{(- 6.65685329)}^{\frac{1}{2}}}$

Step 6.2.2.3

Rewrite $- 6.65685329$ as ${(2.58008784 i)}^{2}$ .

$f' (- 3.828427) = \frac{- 21.31370658}{{({(2.58008784 i)}^{2})}^{\frac{1}{2}}}$

Step 6.2.2.4

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

$f' (- 3.828427) = \frac{- 21.31370658}{{(2.58008784 i)}^{2 (\frac{1}{2})}}$

Step 6.2.2.5

Cancel the common factor of $2$ .

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Step 6.2.2.5.1

Cancel the common factor.

$f' (- 3.828427) = \frac{- 21.31370658}{{(2.58008784 i)}^{2 (\frac{1}{2})}}$

Step 6.2.2.5.2

Rewrite the expression.

$f' (- 3.828427) = \frac{- 21.31370658}{2.58008784 i}$

Step 6.2.2.6

Simplify.

$f' (- 3.828427) = \frac{- 21.31370658}{2.58008784 i}$

Step 6.2.3

Multiply the numerator and denominator of $\frac{- 21.31370658}{2.58008784 i}$ by the conjugate of $2.58008784 i$ to make the denominator real.

$f' (- 3.828427) = \frac{- 21.31370658}{2.58008784 i} \cdot \frac{i}{i}$

Step 6.2.4

Multiply.

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Step 6.2.4.1

Combine.

$f' (- 3.828427) = \frac{- 21.31370658 i}{2.58008784 i i}$

Step 6.2.4.2

Simplify the denominator.

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Step 6.2.4.2.1

Add parentheses.

$f' (- 3.828427) = \frac{- 21.31370658 i}{2.58008784 (i i)}$

Step 6.2.4.2.2

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

$f' (- 3.828427) = \frac{- 21.31370658 i}{2.58008784 (i i)}$

Step 6.2.4.2.3

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

$f' (- 3.828427) = \frac{- 21.31370658 i}{2.58008784 (i i)}$

Step 6.2.4.2.4

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

$f' (- 3.828427) = \frac{- 21.31370658 i}{2.58008784 i^{1 + 1}}$

Step 6.2.4.2.5

Add $1$ and $1$ .

$f' (- 3.828427) = \frac{- 21.31370658 i}{2.58008784 i^{2}}$

Step 6.2.4.2.6

Rewrite $i^{2}$ as $- 1$ .

$f' (- 3.828427) = \frac{- 21.31370658 i}{2.58008784 \cdot - 1}$

Step 6.2.5

Multiply $2.58008784$ by $- 1$ .

$f' (- 3.828427) = \frac{- 21.31370658 i}{- 2.58008784}$

Step 6.2.6

Dividing two negative values results in a positive value.

$f' (- 3.828427) = \frac{(21.31370658) i}{2.58008784}$

Step 6.2.7

Factor $21.31370658$ out of $21.31370658 i$ .

$f' (- 3.828427) = \frac{21.31370658 (i)}{2.58008784}$

Step 6.2.8

Factor $2.58008784$ out of $2.58008784$ .

$f' (- 3.828427) = \frac{21.31370658 (i)}{2.58008784 (1)}$

Step 6.2.9

Separate fractions.

$f' (- 3.828427) = \frac{21.31370658}{2.58008784} \cdot \frac{i}{1}$

Step 6.2.10

Divide $21.31370658$ by $2.58008784$ .

$f' (- 3.828427) = 8.26084531 (\frac{i}{1})$

Step 6.2.11

Divide $i$ by $1$ .

$f' (- 3.828427) = 8.26084531 i$

Step 6.2.12

The final answer is $8.26084531 i$ .

$8.26084531 i$

Step 6.3

At $x = - 3.828427$ the derivative is $8.26084531 i$ . Since this contains an imaginary number, the function does not exist on $(- \infty, - 2 \sqrt{2})$ .

Function is not real on $(- \infty, - 2 \sqrt{2})$ since $f' (x)$ is imaginary

Step 7

Substitute a value from the interval $(- 2.828427, - 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 2.4142135$ in the expression.

$f' (- 2.4142135) = \frac{- 2 {(- 2.4142135)}^{2} + 8}{{(- {(- 2.4142135)}^{2} + 8)}^{\frac{1}{2}}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Raise $- 2.4142135$ to the power of $2$ .

$f' (- 2.4142135) = \frac{- 2 \cdot 5.82842682 + 8}{{(- {(- 2.4142135)}^{2} + 8)}^{\frac{1}{2}}}$

Step 7.2.1.2

Multiply $- 2$ by $5.82842682$ .

$f' (- 2.4142135) = \frac{- 11.65685364 + 8}{{(- {(- 2.4142135)}^{2} + 8)}^{\frac{1}{2}}}$

Step 7.2.1.3

Add $- 11.65685364$ and $8$ .

$f' (- 2.4142135) = \frac{- 3.65685364}{{(- {(- 2.4142135)}^{2} + 8)}^{\frac{1}{2}}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Simplify each term.

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Step 7.2.2.1.1

Raise $- 2.4142135$ to the power of $2$ .

$f' (- 2.4142135) = \frac{- 3.65685364}{{(- 1 \cdot 5.82842682 + 8)}^{\frac{1}{2}}}$

Step 7.2.2.1.2

Multiply $- 1$ by $5.82842682$ .

$f' (- 2.4142135) = \frac{- 3.65685364}{{(- 5.82842682 + 8)}^{\frac{1}{2}}}$

Step 7.2.2.2

Add $- 5.82842682$ and $8$ .

$f' (- 2.4142135) = \frac{- 3.65685364}{{2.17157317}^{\frac{1}{2}}}$

Step 7.2.2.3

Rewrite $2.17157317$ as ${1.47362586}^{2}$ .

$f' (- 2.4142135) = \frac{- 3.65685364}{{({1.47362586}^{2})}^{\frac{1}{2}}}$

Step 7.2.2.4

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

$f' (- 2.4142135) = \frac{- 3.65685364}{{1.47362586}^{2 (\frac{1}{2})}}$

Step 7.2.2.5

Cancel the common factor of $2$ .

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Step 7.2.2.5.1

Cancel the common factor.

$f' (- 2.4142135) = \frac{- 3.65685364}{{1.47362586}^{2 (\frac{1}{2})}}$

Step 7.2.2.5.2

Rewrite the expression.

$f' (- 2.4142135) = \frac{- 3.65685364}{1.47362586}$

Step 7.2.2.6

Evaluate the exponent.

$f' (- 2.4142135) = \frac{- 3.65685364}{1.47362586}$

Step 7.2.3

Divide $- 3.65685364$ by $1.47362586$ .

$f' (- 2.4142135) = - 2.48153465$

Step 7.2.4

The final answer is $- 2.48153465$ .

$- 2.48153465$

Step 7.3

At $x = - 2.4142135$ the derivative is $- 2.48153465$ . Since this is negative, the function is decreasing on $(- 2.828427, - 2)$ .

Decreasing on $(- 2 \sqrt{2}, - 2)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(- 2, 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

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

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

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

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

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

Step 8.2.1.2

Multiply $- 2$ by $0$ .

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

Step 8.2.1.3

Add $0$ and $8$ .

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

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Simplify each term.

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Step 8.2.2.1.1

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

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

Step 8.2.2.1.2

Multiply $- 1$ by $0$ .

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

Step 8.2.2.2

Add $0$ and $8$ .

$f' (0) = \frac{8}{8^{\frac{1}{2}}}$

Step 8.2.3

Move $8^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 8.2.4

Multiply $8$ by $8^{- \frac{1}{2}}$ by adding the exponents.

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Step 8.2.4.1

Multiply $8$ by $8^{- \frac{1}{2}}$ .

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Step 8.2.4.1.1

Raise $8$ to the power of $1$ .

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

Step 8.2.4.1.2

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

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

Step 8.2.4.2

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

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

Step 8.2.4.3

Combine the numerators over the common denominator.

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

Step 8.2.4.4

Subtract $1$ from $2$ .

$f' (0) = 8^{\frac{1}{2}}$

Step 8.2.5

The final answer is $8^{\frac{1}{2}}$ .

$8^{\frac{1}{2}}$

Step 8.3

At $x = 0$ the derivative is $8^{\frac{1}{2}}$ . Since this is positive, the function is increasing on $(- 2, 2)$ .

Increasing on $(- 2, 2)$ since $f' (x) > 0$

Step 9

Substitute a value from the interval $(2, 2 \sqrt{2})$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $2.4142135$ in the expression.

$f' (2.4142135) = \frac{- 2 {(2.4142135)}^{2} + 8}{{(- {(2.4142135)}^{2} + 8)}^{\frac{1}{2}}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify the numerator.

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Step 9.2.1.1

Raise $2.4142135$ to the power of $2$ .

$f' (2.4142135) = \frac{- 2 \cdot 5.82842682 + 8}{{(- {2.4142135}^{2} + 8)}^{\frac{1}{2}}}$

Step 9.2.1.2

Multiply $- 2$ by $5.82842682$ .

$f' (2.4142135) = \frac{- 11.65685364 + 8}{{(- {2.4142135}^{2} + 8)}^{\frac{1}{2}}}$

Step 9.2.1.3

Add $- 11.65685364$ and $8$ .

$f' (2.4142135) = \frac{- 3.65685364}{{(- {2.4142135}^{2} + 8)}^{\frac{1}{2}}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Simplify each term.

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Step 9.2.2.1.1

Raise $2.4142135$ to the power of $2$ .

$f' (2.4142135) = \frac{- 3.65685364}{{(- 1 \cdot 5.82842682 + 8)}^{\frac{1}{2}}}$

Step 9.2.2.1.2

Multiply $- 1$ by $5.82842682$ .

$f' (2.4142135) = \frac{- 3.65685364}{{(- 5.82842682 + 8)}^{\frac{1}{2}}}$

Step 9.2.2.2

Add $- 5.82842682$ and $8$ .

$f' (2.4142135) = \frac{- 3.65685364}{{2.17157317}^{\frac{1}{2}}}$

Step 9.2.2.3

Rewrite $2.17157317$ as ${1.47362586}^{2}$ .

$f' (2.4142135) = \frac{- 3.65685364}{{({1.47362586}^{2})}^{\frac{1}{2}}}$

Step 9.2.2.4

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

$f' (2.4142135) = \frac{- 3.65685364}{{1.47362586}^{2 (\frac{1}{2})}}$

Step 9.2.2.5

Cancel the common factor of $2$ .

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Step 9.2.2.5.1

Cancel the common factor.

$f' (2.4142135) = \frac{- 3.65685364}{{1.47362586}^{2 (\frac{1}{2})}}$

Step 9.2.2.5.2

Rewrite the expression.

$f' (2.4142135) = \frac{- 3.65685364}{1.47362586}$

Step 9.2.2.6

Evaluate the exponent.

$f' (2.4142135) = \frac{- 3.65685364}{1.47362586}$

Step 9.2.3

Divide $- 3.65685364$ by $1.47362586$ .

$f' (2.4142135) = - 2.48153465$

Step 9.2.4

The final answer is $- 2.48153465$ .

$- 2.48153465$

Step 9.3

At $x = 2.4142135$ the derivative is $- 2.48153465$ . Since this is negative, the function is decreasing on $(2, 2 \sqrt{2})$ .

Decreasing on $(2, 2 \sqrt{2})$ since $f' (x) < 0$

Step 10

Substitute a value from the interval $(2 \sqrt{2}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 10.1

Replace the variable $x$ with $3.828427$ in the expression.

$f' (3.828427) = \frac{- 2 {(3.828427)}^{2} + 8}{{(- {(3.828427)}^{2} + 8)}^{\frac{1}{2}}}$

Step 10.2

Simplify the result.

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Step 10.2.1

Simplify the numerator.

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Step 10.2.1.1

Raise $3.828427$ to the power of $2$ .

$f' (3.828427) = \frac{- 2 \cdot 14.65685329 + 8}{{(- {3.828427}^{2} + 8)}^{\frac{1}{2}}}$

Step 10.2.1.2

Multiply $- 2$ by $14.65685329$ .

$f' (3.828427) = \frac{- 29.31370658 + 8}{{(- {3.828427}^{2} + 8)}^{\frac{1}{2}}}$

Step 10.2.1.3

Add $- 29.31370658$ and $8$ .

$f' (3.828427) = \frac{- 21.31370658}{{(- {3.828427}^{2} + 8)}^{\frac{1}{2}}}$

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Simplify each term.

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Step 10.2.2.1.1

Raise $3.828427$ to the power of $2$ .

$f' (3.828427) = \frac{- 21.31370658}{{(- 1 \cdot 14.65685329 + 8)}^{\frac{1}{2}}}$

Step 10.2.2.1.2

Multiply $- 1$ by $14.65685329$ .

$f' (3.828427) = \frac{- 21.31370658}{{(- 14.65685329 + 8)}^{\frac{1}{2}}}$

Step 10.2.2.2

Add $- 14.65685329$ and $8$ .

$f' (3.828427) = \frac{- 21.31370658}{{(- 6.65685329)}^{\frac{1}{2}}}$

Step 10.2.2.3

Rewrite $- 6.65685329$ as ${(2.58008784 i)}^{2}$ .

$f' (3.828427) = \frac{- 21.31370658}{{({(2.58008784 i)}^{2})}^{\frac{1}{2}}}$

Step 10.2.2.4

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

$f' (3.828427) = \frac{- 21.31370658}{{(2.58008784 i)}^{2 (\frac{1}{2})}}$

Step 10.2.2.5

Cancel the common factor of $2$ .

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Step 10.2.2.5.1

Cancel the common factor.

$f' (3.828427) = \frac{- 21.31370658}{{(2.58008784 i)}^{2 (\frac{1}{2})}}$

Step 10.2.2.5.2

Rewrite the expression.

$f' (3.828427) = \frac{- 21.31370658}{2.58008784 i}$

Step 10.2.2.6

Simplify.

$f' (3.828427) = \frac{- 21.31370658}{2.58008784 i}$

Step 10.2.3

Multiply the numerator and denominator of $\frac{- 21.31370658}{2.58008784 i}$ by the conjugate of $2.58008784 i$ to make the denominator real.

$f' (3.828427) = \frac{- 21.31370658}{2.58008784 i} \cdot \frac{i}{i}$

Step 10.2.4

Multiply.

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Step 10.2.4.1

Combine.

$f' (3.828427) = \frac{- 21.31370658 i}{2.58008784 i i}$

Step 10.2.4.2

Simplify the denominator.

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Step 10.2.4.2.1

Add parentheses.

$f' (3.828427) = \frac{- 21.31370658 i}{2.58008784 (i i)}$

Step 10.2.4.2.2

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

$f' (3.828427) = \frac{- 21.31370658 i}{2.58008784 (i i)}$

Step 10.2.4.2.3

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

$f' (3.828427) = \frac{- 21.31370658 i}{2.58008784 (i i)}$

Step 10.2.4.2.4

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

$f' (3.828427) = \frac{- 21.31370658 i}{2.58008784 i^{1 + 1}}$

Step 10.2.4.2.5

Add $1$ and $1$ .

$f' (3.828427) = \frac{- 21.31370658 i}{2.58008784 i^{2}}$

Step 10.2.4.2.6

Rewrite $i^{2}$ as $- 1$ .

$f' (3.828427) = \frac{- 21.31370658 i}{2.58008784 \cdot - 1}$

Step 10.2.5

Multiply $2.58008784$ by $- 1$ .

$f' (3.828427) = \frac{- 21.31370658 i}{- 2.58008784}$

Step 10.2.6

Dividing two negative values results in a positive value.

$f' (3.828427) = \frac{(21.31370658) i}{2.58008784}$

Step 10.2.7

Factor $21.31370658$ out of $21.31370658 i$ .

$f' (3.828427) = \frac{21.31370658 (i)}{2.58008784}$

Step 10.2.8

Factor $2.58008784$ out of $2.58008784$ .

$f' (3.828427) = \frac{21.31370658 (i)}{2.58008784 (1)}$

Step 10.2.9

Separate fractions.

$f' (3.828427) = \frac{21.31370658}{2.58008784} \cdot \frac{i}{1}$

Step 10.2.10

Divide $21.31370658$ by $2.58008784$ .

$f' (3.828427) = 8.26084531 (\frac{i}{1})$

Step 10.2.11

Divide $i$ by $1$ .

$f' (3.828427) = 8.26084531 i$

Step 10.2.12

The final answer is $8.26084531 i$ .

$8.26084531 i$

Step 10.3

At $x = 3.828427$ the derivative is $8.26084531 i$ . Since this contains an imaginary number, the function does not exist on $(2 \sqrt{2}, \infty)$ .

Function is not real on $(2 \sqrt{2}, \infty)$ since $f' (x)$ is imaginary

Step 11

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- 2, 2)$

Decreasing on: $(- 2 \sqrt{2}, - 2), (2, 2 \sqrt{2})$

Step 12