Calculus Examples

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

$(1, 1)$

Step 1

Find the first derivative and evaluate at

$x = 1$ and

$y = 1$ 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

$\sqrt{2 - x^{2}}$ as

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

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

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

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

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

Step 1.3

Multiply the exponents in

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

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Step 1.3.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.3.2

Cancel the common factor of

$2$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.4

Simplify.

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

Step 1.5

The derivative of

$| x |$ with respect to

$x$ is

$\frac{x}{| x |}$ .

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

Step 1.6

Combine

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

$\frac{x}{| x |}$ .

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

Step 1.7

Multiply

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

$\frac{| x |}{| x |}$ .

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

Step 1.8

Combine.

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

Step 1.9

Apply the distributive property.

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

Step 1.10

Cancel the common factor of

$| x |$ .

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Step 1.10.1

Cancel the common factor.

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

Step 1.10.2

Rewrite the expression.

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

Step 1.11

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

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

Step 1.12

Raise

$x$ to the power of

$1$ .

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

Step 1.13

Raise

$x$ to the power of

$1$ .

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

Step 1.14

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.15

Add

$1$ and

$1$ .

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

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

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

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Step 1.16.1

To apply the Chain Rule, set

$u$ as

$2 - x^{2}$ .

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

Step 1.16.2

Differentiate using the Power Rule which states that

$\frac{d}{d u} [u^{n}]$ is

$n u^{n - 1}$ where

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

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

Step 1.16.3

Replace all occurrences of

$u$ with

$2 - x^{2}$ .

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

Step 1.17

To write

$- 1$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 1.18

Combine

$- 1$ and

$\frac{2}{2}$ .

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

Step 1.19

Combine the numerators over the common denominator.

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

Step 1.20

Simplify the numerator.

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Step 1.20.1

Multiply

$- 1$ by

$2$ .

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

Step 1.20.2

Subtract

$2$ from

$1$ .

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

Step 1.21

Combine fractions.

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Step 1.21.1

Move the negative in front of the fraction.

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

Step 1.21.2

Combine

$\frac{1}{2}$ and

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

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

Step 1.21.3

Move

${(2 - x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule

$b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.21.4

Combine

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

$| x^{2} |$ .

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

Step 1.22

By the Sum Rule, the derivative of

$2 - x^{2}$ with respect to

$x$ is

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

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

Step 1.23

Since

$2$ is constant with respect to

$x$ , the derivative of

$2$ with respect to

$x$ is

$0$ .

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

Step 1.24

Add

$0$ and

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

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

Step 1.25

Since

$- 1$ is constant with respect to

$x$ , the derivative of

$- x^{2}$ with respect to

$x$ is

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

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

Step 1.26

Multiply.

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Step 1.26.1

Multiply

$- 1$ by

$- 1$ .

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

Step 1.26.2

Multiply

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

$1$ .

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

Step 1.27

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

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

Step 1.28

Simplify terms.

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Step 1.28.1

Combine

$2$ and

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

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

Step 1.28.2

Combine

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

$x$ .

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

Step 1.28.3

Cancel the common factor.

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

Step 1.28.4

Rewrite the expression.

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

Step 1.28.5

Reorder

$2$ and

$- x^{2}$ .

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

Step 1.29

To write

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

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

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

Step 1.30

Combine the numerators over the common denominator.

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

Step 1.31

Multiply

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

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

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Step 1.31.1

Move

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

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

Step 1.31.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.31.3

Combine the numerators over the common denominator.

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

Step 1.31.4

Add

$1$ and

$1$ .

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

Step 1.31.5

Divide

$2$ by

$2$ .

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

Step 1.32

Simplify

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

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

Step 1.33

Rewrite

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

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

Step 1.34

Multiply

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

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

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

Step 1.35

Reorder terms.

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

Step 1.36

Multiply

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

$- x^{2} + 2$ by adding the exponents.

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Step 1.36.1

Move

$- x^{2} + 2$ .

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

Step 1.36.2

Multiply

$- x^{2} + 2$ by

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

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Step 1.36.2.1

Raise

$- x^{2} + 2$ to the power of

$1$ .

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

Step 1.36.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.36.3

Write

$1$ as a fraction with a common denominator.

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

Step 1.36.4

Combine the numerators over the common denominator.

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

Step 1.36.5

Add

$2$ and

$1$ .

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

Step 1.37

Simplify.

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Step 1.37.1

Apply the distributive property.

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

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.

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

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

$x^{2}$ .

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

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

$1$ .

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

Step 1.37.2.1.2.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.37.2.1.2.3

Add

$2$ and

$1$ .

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

Step 1.37.2.1.3

Move

$2$ to the left of

$x$ .

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

Step 1.37.2.1.4

Remove non-negative terms from the absolute value.

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

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

$1$ .

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

Step 1.37.2.1.5.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.37.2.1.5.2

Add

$1$ and

$2$ .

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

Step 1.37.2.2

Combine the opposite terms in

$- x^{3} + 2 x + x^{3}$ .

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Step 1.37.2.2.1

Add

$- x^{3}$ and

$x^{3}$ .

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

Step 1.37.2.2.2

Add

$2 x$ and

$0$ .

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

Step 1.38

Evaluate the derivative at

$x = 1$ .

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

Step 1.39

Simplify.

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Step 1.39.1

Multiply

$2$ by

$1$ .

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

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

One to any power is one.

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

Step 1.39.2.1.2

Multiply

$- 1$ by

$1$ .

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

Step 1.39.2.2

Add

$- 1$ and

$2$ .

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

Step 1.39.2.3

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

$0$ and

$1$ is

$1$ .

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

Step 1.39.2.4

One to any power is one.

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

Step 1.39.3

Simplify the expression.

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Step 1.39.3.1

Multiply

$1$ by

$1$ .

$\frac{2}{1}$

Step 1.39.3.2

Divide

$2$ by

$1$ .

$2$

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

$2$ and a given point

$(1, 1)$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = 2 \cdot (x - (1))$

Step 2.2

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

$y - 1 = 2 \cdot (x - 1)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Simplify

$2 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + 2 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = 2 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = 2 x + 2 \cdot - 1$

Step 2.3.1.4

Multiply

$2$ by

$- 1$ .

$y - 1 = 2 x - 2$

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

$1$ to both sides of the equation.

$y = 2 x - 2 + 1$

Step 2.3.2.2

Add

$- 2$ and

$1$ .

$y = 2 x - 1$

Step 3