Calculus Examples

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

Step 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 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 3

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

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Step 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 3.2

Cancel the common factor of $2$ .

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Step 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 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 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 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $1$ .

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

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Multiply.

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Step 16.1

Multiply $- 1$ by $- 1$ .

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

Step 16.2

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

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

Step 17

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

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

Step 18

Combine fractions.

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Step 18.1

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

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

Step 18.2

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

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

Step 19

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

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

Step 20

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

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

Step 21

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

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

Step 22

Add $1$ and $1$ .

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

Step 23

Cancel the common factor.

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

Step 24

Rewrite the expression.

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

Step 25

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

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

Step 26

Combine the numerators over the common denominator.

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

Step 27

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

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Step 27.1

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

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

Step 27.2

Combine the numerators over the common denominator.

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

Step 27.3

Add $1$ and $1$ .

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

Step 27.4

Divide $2$ by $2$ .

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

Step 28

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

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

Step 29

Add $- x^{2}$ and $x^{2}$ .

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

Step 30

Add $2$ and $0$ .

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

Step 31

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

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

Step 32

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

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

Step 33

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

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Step 33.1

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

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Step 33.1.1

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

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

Step 33.1.2

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

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

Step 33.2

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

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

Step 33.3

Combine the numerators over the common denominator.

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

Step 33.4

Add $1$ and $2$ .

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

Step 34

Reorder terms.

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