Calculus Examples

$y = \frac{| x |}{\sqrt{5 - x^{2}}}$ , $(2, 2)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 2$ to find the slope of the tangent line.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 - x^{2}}$ as ${(5 - x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{| x |}{{(5 - x^{2})}^{\frac{1}{2}}}]$

Step 1.2

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [| x |] - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{{({(5 - x^{2})}^{\frac{1}{2}})}^{2}}$

Step 1.3

Multiply the exponents in ${({(5 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 1.3.1

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [| x |] - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{{(5 - x^{2})}^{\frac{1}{2} \cdot 2}}$

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

$\frac{{(5 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [| x |] - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{{(5 - x^{2})}^{\frac{1}{2} \cdot 2}}$

Step 1.3.2.2

Rewrite the expression.

$\frac{{(5 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [| x |] - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{{(5 - x^{2})}^{1}}$

Step 1.4

Simplify.

$\frac{{(5 - x^{2})}^{\frac{1}{2}} \frac{d}{d x} [| x |] - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{5 - x^{2}}$

Step 1.5

The derivative of $| x |$ with respect to $x$ is $\frac{x}{| x |}$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} \frac{x}{| x |} - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{5 - x^{2}}$

Step 1.6

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

$\frac{\frac{{(5 - x^{2})}^{\frac{1}{2}} x}{| x |} - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{5 - x^{2}}$

Step 1.7

Multiply $\frac{\frac{{(5 - x^{2})}^{\frac{1}{2}} x}{| x |} - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{5 - x^{2}}$ by $\frac{| x |}{| x |}$ .

$\frac{| x |}{| x |} \cdot \frac{\frac{{(5 - x^{2})}^{\frac{1}{2}} x}{| x |} - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{5 - x^{2}}$

Step 1.8

Combine.

$\frac{| x | (\frac{{(5 - x^{2})}^{\frac{1}{2}} x}{| x |} - | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}])}{| x | (5 - x^{2})}$

Step 1.9

Apply the distributive property.

$\frac{| x | \frac{{(5 - x^{2})}^{\frac{1}{2}} x}{| x |} + | x | (- | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}])}{| x | (5 - x^{2})}$

Step 1.10

Cancel the common factor of $| x |$ .

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Step 1.10.1

Cancel the common factor.

$\frac{| x | \frac{{(5 - x^{2})}^{\frac{1}{2}} x}{| x |} + | x | (- | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}])}{| x | (5 - x^{2})}$

Step 1.10.2

Rewrite the expression.

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x + | x | (- | x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}])}{| x | (5 - x^{2})}$

Step 1.11

To multiply absolute values, multiply the terms inside each absolute value.

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x \cdot x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{| x | (5 - x^{2})}$

Step 1.12

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{1} x | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{| x | (5 - x^{2})}$

Step 1.13

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{1} x^{1} | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{| x | (5 - x^{2})}$

Step 1.14

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{1 + 1} | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{| x | (5 - x^{2})}$

Step 1.15

Add $1$ and $1$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | \frac{d}{d x} [{(5 - x^{2})}^{\frac{1}{2}}]}{| x | (5 - x^{2})}$

Step 1.16

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 $g (x) = 5 - x^{2}$ .

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Step 1.16.1

To apply the Chain Rule, set $u$ as $5 - x^{2}$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.16.2

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.16.3

Replace all occurrences of $u$ with $5 - x^{2}$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{1}{2} {(5 - x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.17

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{1}{2} {(5 - x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.18

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{1}{2} {(5 - x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.19

Combine the numerators over the common denominator.

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{1}{2} {(5 - x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.20

Simplify the numerator.

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Step 1.20.1

Multiply $- 1$ by $2$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{1}{2} {(5 - x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.20.2

Subtract $2$ from $1$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{1}{2} {(5 - x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.21

Combine fractions.

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Step 1.21.1

Move the negative in front of the fraction.

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{1}{2} {(5 - x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.21.2

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{{(5 - x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.21.3

Move ${(5 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - | x^{2} | (\frac{1}{2 {(5 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [5 - x^{2}])}{| x | (5 - x^{2})}$

Step 1.21.4

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - \frac{| x^{2} |}{2 {(5 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [5 - x^{2}]}{| x | (5 - x^{2})}$

Step 1.22

By the Sum Rule, the derivative of $5 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- x^{2}]$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - \frac{| x^{2} |}{2 {(5 - x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [5] + \frac{d}{d x} [- x^{2}])}{| x | (5 - x^{2})}$

Step 1.23

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - \frac{| x^{2} |}{2 {(5 - x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [- x^{2}])}{| x | (5 - x^{2})}$

Step 1.24

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - \frac{| x^{2} |}{2 {(5 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [- x^{2}]}{| x | (5 - x^{2})}$

Step 1.25

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x - \frac{| x^{2} |}{2 {(5 - x^{2})}^{\frac{1}{2}}} (- \frac{d}{d x} [x^{2}])}{| x | (5 - x^{2})}$

Step 1.26

Multiply.

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Step 1.26.1

Multiply $- 1$ by $- 1$ .

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x + 1 \frac{| x^{2} |}{2 {(5 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]}{| x | (5 - x^{2})}$

Step 1.26.2

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x + \frac{| x^{2} |}{2 {(5 - x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]}{| x | (5 - x^{2})}$

Step 1.27

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x + \frac{| x^{2} |}{2 {(5 - x^{2})}^{\frac{1}{2}}} (2 x)}{| x | (5 - x^{2})}$

Step 1.28

Simplify terms.

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Step 1.28.1

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x + \frac{2 | x^{2} |}{2 {(5 - x^{2})}^{\frac{1}{2}}} x}{| x | (5 - x^{2})}$

Step 1.28.2

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

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x + \frac{2 | x^{2} | x}{2 {(5 - x^{2})}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.28.3

Cancel the common factor.

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x + \frac{2 | x^{2} | x}{2 {(5 - x^{2})}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.28.4

Rewrite the expression.

$\frac{{(5 - x^{2})}^{\frac{1}{2}} x + \frac{| x^{2} | x}{{(5 - x^{2})}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.28.5

Reorder $5$ and $- x^{2}$ .

$\frac{x {(- x^{2} + 5)}^{\frac{1}{2}} + \frac{x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.29

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

$\frac{\frac{x {(- x^{2} + 5)}^{\frac{1}{2}} {(- x^{2} + 5)}^{\frac{1}{2}}}{{(- x^{2} + 5)}^{\frac{1}{2}}} + \frac{x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.30

Combine the numerators over the common denominator.

$\frac{\frac{x {(- x^{2} + 5)}^{\frac{1}{2}} {(- x^{2} + 5)}^{\frac{1}{2}} + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.31

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

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Step 1.31.1

Move ${(- x^{2} + 5)}^{\frac{1}{2}}$ .

$\frac{\frac{x ({(- x^{2} + 5)}^{\frac{1}{2}} {(- x^{2} + 5)}^{\frac{1}{2}}) + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.31.2

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

$\frac{\frac{x {(- x^{2} + 5)}^{\frac{1}{2} + \frac{1}{2}} + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.31.3

Combine the numerators over the common denominator.

$\frac{\frac{x {(- x^{2} + 5)}^{\frac{1 + 1}{2}} + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.31.4

Add $1$ and $1$ .

$\frac{\frac{x {(- x^{2} + 5)}^{\frac{2}{2}} + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.31.5

Divide $2$ by $2$ .

$\frac{\frac{x {(- x^{2} + 5)}^{1} + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.32

Simplify $x {(- x^{2} + 5)}^{1}$ .

$\frac{\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$

Step 1.33

Rewrite $\frac{\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}}}{| x | (5 - x^{2})}$ as a product.

$\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}}} \cdot \frac{1}{| x | (5 - x^{2})}$

Step 1.34

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

$\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}} (| x | (5 - x^{2}))}$

Step 1.35

Reorder terms.

$\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{1}{2}} (| x | (- x^{2} + 5))}$

Step 1.36

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

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Step 1.36.1

Move $- x^{2} + 5$ .

$\frac{x (- x^{2} + 5) + x | x^{2} |}{(- x^{2} + 5) {(- x^{2} + 5)}^{\frac{1}{2}} | x |}$

Step 1.36.2

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

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Step 1.36.2.1

Raise $- x^{2} + 5$ to the power of $1$ .

$\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{1} {(- x^{2} + 5)}^{\frac{1}{2}} | x |}$

Step 1.36.2.2

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

$\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{1 + \frac{1}{2}} | x |}$

Step 1.36.3

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

$\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{2}{2} + \frac{1}{2}} | x |}$

Step 1.36.4

Combine the numerators over the common denominator.

$\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{2 + 1}{2}} | x |}$

Step 1.36.5

Add $2$ and $1$ .

$\frac{x (- x^{2} + 5) + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37

Simplify.

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Step 1.37.1

Apply the distributive property.

$\frac{x (- x^{2}) + x \cdot 5 + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2

Simplify the numerator.

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Step 1.37.2.1

Simplify each term.

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Step 1.37.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{- x \cdot x^{2} + x \cdot 5 + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.1.2

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

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Step 1.37.2.1.2.1

Move $x^{2}$ .

$\frac{- (x^{2} x) + x \cdot 5 + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.1.2.2

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

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Step 1.37.2.1.2.2.1

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

$\frac{- (x^{2} x^{1}) + x \cdot 5 + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.1.2.2.2

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

$\frac{- x^{2 + 1} + x \cdot 5 + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.1.2.3

Add $2$ and $1$ .

$\frac{- x^{3} + x \cdot 5 + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.1.3

Move $5$ to the left of $x$ .

$\frac{- x^{3} + 5 \cdot x + x | x^{2} |}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.1.4

Remove non-negative terms from the absolute value.

$\frac{- x^{3} + 5 x + x \cdot x^{2}}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.1.5

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

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Step 1.37.2.1.5.1

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

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Step 1.37.2.1.5.1.1

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

$\frac{- x^{3} + 5 x + x^{1} x^{2}}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.1.5.1.2

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

$\frac{- x^{3} + 5 x + x^{1 + 2}}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.1.5.2

Add $1$ and $2$ .

$\frac{- x^{3} + 5 x + x^{3}}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.2

Combine the opposite terms in $- x^{3} + 5 x + x^{3}$ .

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Step 1.37.2.2.1

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

$\frac{5 x + 0}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.37.2.2.2

Add $5 x$ and $0$ .

$\frac{5 x}{{(- x^{2} + 5)}^{\frac{3}{2}} | x |}$

Step 1.38

Evaluate the derivative at $x = 2$ .

$\frac{5 (2)}{{(- {(2)}^{2} + 5)}^{\frac{3}{2}} | 2 |}$

Step 1.39

Simplify.

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Step 1.39.1

Multiply $5$ by $2$ .

$\frac{10}{{(- 2^{2} + 5)}^{\frac{3}{2}} | 2 |}$

Step 1.39.2

Simplify the denominator.

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Step 1.39.2.1

Simplify each term.

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Step 1.39.2.1.1

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

$\frac{10}{{(- 1 \cdot 4 + 5)}^{\frac{3}{2}} | 2 |}$

Step 1.39.2.1.2

Multiply $- 1$ by $4$ .

$\frac{10}{{(- 4 + 5)}^{\frac{3}{2}} | 2 |}$

Step 1.39.2.2

Add $- 4$ and $5$ .

$\frac{10}{1^{\frac{3}{2}} | 2 |}$

Step 1.39.2.3

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

$\frac{10}{1^{\frac{3}{2}} \cdot 2}$

Step 1.39.2.4

One to any power is one.

$\frac{10}{1 \cdot 2}$

Step 1.39.3

Simplify the expression.

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Step 1.39.3.1

Multiply $2$ by $1$ .

$\frac{10}{2}$

Step 1.39.3.2

Divide $10$ by $2$ .

$5$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $5$ and a given point $(2, 2)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = 5 \cdot (x - (2))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - 2 = 5 \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $5 \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 + 5 \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 2 = 5 \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 2 = 5 x + 5 \cdot - 2$

Step 2.3.1.4

Multiply $5$ by $- 2$ .

$y - 2 = 5 x - 10$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.2.1

Add $2$ to both sides of the equation.

$y = 5 x - 10 + 2$

Step 2.3.2.2

Add $- 10$ and $2$ .

$y = 5 x - 8$

Step 3