Calculus Examples

$f (x) = \sin (\arctan (5 x))$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Draw a triangle in the plane with vertices $(1, 5 x)$ , $(1, 0)$ , and the origin. Then $\arctan (5 x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, 5 x)$ . Therefore, $\sin (\arctan (5 x))$ is $\frac{5 x}{\sqrt{1 + {(5 x)}^{2}}}$ .

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

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Factor $5$ out of $5 x$ .

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

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Apply the product rule to $5 (x)$ .

$\frac{d}{d x} [\frac{5 x}{\sqrt{1 + 5^{2} x^{2}}}]$

Step 1.2.2.2

Raise $5$ to the power of $2$ .

$\frac{d}{d x} [\frac{5 x}{\sqrt{1 + 25 x^{2}}}]$

Step 1.2.2.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + 25 x^{2}}$ as ${(1 + 25 x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{5 x}{{(1 + 25 x^{2})}^{\frac{1}{2}}}]$

Step 1.3

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5 x}{{(1 + 25 x^{2})}^{\frac{1}{2}}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{x}{{(1 + 25 x^{2})}^{\frac{1}{2}}}]$ .

$5 \frac{d}{d x} [\frac{x}{{(1 + 25 x^{2})}^{\frac{1}{2}}}]$

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 $g (x) = {(1 + 25 x^{2})}^{\frac{1}{2}}$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(1 + 25 x^{2})}^{\frac{1}{2}}]}{{({(1 + 25 x^{2})}^{\frac{1}{2}})}^{2}}$

Step 3

Multiply the exponents in ${({(1 + 25 x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.1

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

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(1 + 25 x^{2})}^{\frac{1}{2}}]}{{(1 + 25 x^{2})}^{\frac{1}{2} \cdot 2}}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(1 + 25 x^{2})}^{\frac{1}{2}}]}{{(1 + 25 x^{2})}^{\frac{1}{2} \cdot 2}}$

Step 3.2.2

Rewrite the expression.

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(1 + 25 x^{2})}^{\frac{1}{2}}]}{{(1 + 25 x^{2})}^{1}}$

Step 4

Simplify.

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} \frac{d}{d x} [x] - x \frac{d}{d x} [{(1 + 25 x^{2})}^{\frac{1}{2}}]}{1 + 25 x^{2}}$

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 $n = 1$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} \cdot 1 - x \frac{d}{d x} [{(1 + 25 x^{2})}^{\frac{1}{2}}]}{1 + 25 x^{2}}$

Step 5.2

Multiply ${(1 + 25 x^{2})}^{\frac{1}{2}}$ by $1$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x \frac{d}{d x} [{(1 + 25 x^{2})}^{\frac{1}{2}}]}{1 + 25 x^{2}}$

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 $g (x) = 1 + 25 x^{2}$ .

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Step 6.1

To apply the Chain Rule, set $u$ as $1 + 25 x^{2}$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 6.2

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

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 6.3

Replace all occurrences of $u$ with $1 + 25 x^{2}$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} {(1 + 25 x^{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 7

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

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} {(1 + 25 x^{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 8

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

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} {(1 + 25 x^{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 9

Combine the numerators over the common denominator.

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} {(1 + 25 x^{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} {(1 + 25 x^{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 10.2

Subtract $2$ from $1$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} {(1 + 25 x^{2})}^{\frac{- 1}{2}} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 11

Combine fractions.

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Step 11.1

Move the negative in front of the fraction.

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{1}{2} {(1 + 25 x^{2})}^{- \frac{1}{2}} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 11.2

Combine $\frac{1}{2}$ and ${(1 + 25 x^{2})}^{- \frac{1}{2}}$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{{(1 + 25 x^{2})}^{- \frac{1}{2}}}{2} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 11.3

Move ${(1 + 25 x^{2})}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - x (\frac{1}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [1 + 25 x^{2}])}{1 + 25 x^{2}}$

Step 11.4

Combine $\frac{1}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}$ and $x$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [1 + 25 x^{2}]}{1 + 25 x^{2}}$

Step 12

By the Sum Rule, the derivative of $1 + 25 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [25 x^{2}]$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} (\frac{d}{d x} [1] + \frac{d}{d x} [25 x^{2}])}{1 + 25 x^{2}}$

Step 13

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} (0 + \frac{d}{d x} [25 x^{2}])}{1 + 25 x^{2}}$

Step 14

Add $0$ and $\frac{d}{d x} [25 x^{2}]$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [25 x^{2}]}{1 + 25 x^{2}}$

Step 15

Since $25$ is constant with respect to $x$ , the derivative of $25 x^{2}$ with respect to $x$ is $25 \frac{d}{d x} [x^{2}]$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - \frac{x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} (25 \frac{d}{d x} [x^{2}])}{1 + 25 x^{2}}$

Step 16

Combine fractions.

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Step 16.1

Multiply $25$ by $- 1$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - 25 \frac{x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]}{1 + 25 x^{2}}$

Step 16.2

Combine $- 25$ and $\frac{x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{- 25 x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]}{1 + 25 x^{2}}$

Step 16.3

Move the negative in front of the fraction.

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - \frac{25 x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} \frac{d}{d x} [x^{2}]}{1 + 25 x^{2}}$

Step 17

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

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - \frac{25 x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} (2 x)}{1 + 25 x^{2}}$

Step 18

Combine fractions.

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Step 18.1

Multiply $2$ by $- 1$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - 2 \frac{25 x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} x}{1 + 25 x^{2}}$

Step 18.2

Combine $- 2$ and $\frac{25 x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{- 2 (25 x)}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} x}{1 + 25 x^{2}}$

Step 18.3

Multiply $25$ by $- 2$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{- 50 x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}} x}{1 + 25 x^{2}}$

Step 18.4

Combine $\frac{- 50 x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}$ and $x$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{- 50 x \cdot x}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 19

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

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{- 50 (x^{1} x)}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 20

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

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{- 50 (x^{1} x^{1})}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 21

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

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{- 50 x^{1 + 1}}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 22

Add $1$ and $1$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{- 50 x^{2}}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 23

Factor $2$ out of $- 50 x^{2}$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{2 (- 25 x^{2})}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 24

Cancel the common factors.

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Step 24.1

Factor $2$ out of $2 {(1 + 25 x^{2})}^{\frac{1}{2}}$ .

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{2 (- 25 x^{2})}{2 ({(1 + 25 x^{2})}^{\frac{1}{2}})}}{1 + 25 x^{2}}$

Step 24.2

Cancel the common factor.

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{2 (- 25 x^{2})}{2 {(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 24.3

Rewrite the expression.

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} + \frac{- 25 x^{2}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 25

Move the negative in front of the fraction.

$5 \frac{{(1 + 25 x^{2})}^{\frac{1}{2}} - \frac{25 x^{2}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 26

To write ${(1 + 25 x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(1 + 25 x^{2})}^{\frac{1}{2}}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}$ .

$5 \frac{\frac{{(1 + 25 x^{2})}^{\frac{1}{2}} {(1 + 25 x^{2})}^{\frac{1}{2}}}{{(1 + 25 x^{2})}^{\frac{1}{2}}} - \frac{25 x^{2}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 27

Combine the numerators over the common denominator.

$5 \frac{\frac{{(1 + 25 x^{2})}^{\frac{1}{2}} {(1 + 25 x^{2})}^{\frac{1}{2}} - 25 x^{2}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 28

Multiply ${(1 + 25 x^{2})}^{\frac{1}{2}}$ by ${(1 + 25 x^{2})}^{\frac{1}{2}}$ by adding the exponents.

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Step 28.1

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

$5 \frac{\frac{{(1 + 25 x^{2})}^{\frac{1}{2} + \frac{1}{2}} - 25 x^{2}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 28.2

Combine the numerators over the common denominator.

$5 \frac{\frac{{(1 + 25 x^{2})}^{\frac{1 + 1}{2}} - 25 x^{2}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 28.3

Add $1$ and $1$ .

$5 \frac{\frac{{(1 + 25 x^{2})}^{\frac{2}{2}} - 25 x^{2}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 28.4

Divide $2$ by $2$ .

$5 \frac{\frac{{(1 + 25 x^{2})}^{1} - 25 x^{2}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 29

Simplify ${(1 + 25 x^{2})}^{1}$ .

$5 \frac{\frac{1 + 25 x^{2} - 25 x^{2}}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 30

Subtract $25 x^{2}$ from $25 x^{2}$ .

$5 \frac{\frac{1 + 0}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 31

Add $1$ and $0$ .

$5 \frac{\frac{1}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$

Step 32

Rewrite $\frac{\frac{1}{{(1 + 25 x^{2})}^{\frac{1}{2}}}}{1 + 25 x^{2}}$ as a product.

$5 (\frac{1}{{(1 + 25 x^{2})}^{\frac{1}{2}}} \cdot \frac{1}{1 + 25 x^{2}})$

Step 33

Multiply $\frac{1}{{(1 + 25 x^{2})}^{\frac{1}{2}}}$ by $\frac{1}{1 + 25 x^{2}}$ .

$5 \frac{1}{{(1 + 25 x^{2})}^{\frac{1}{2}} (1 + 25 x^{2})}$

Step 34

Multiply ${(1 + 25 x^{2})}^{\frac{1}{2}}$ by $1 + 25 x^{2}$ by adding the exponents.

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Step 34.1

Multiply ${(1 + 25 x^{2})}^{\frac{1}{2}}$ by $1 + 25 x^{2}$ .

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Step 34.1.1

Raise $1 + 25 x^{2}$ to the power of $1$ .

$5 \frac{1}{{(1 + 25 x^{2})}^{\frac{1}{2}} {(1 + 25 x^{2})}^{1}}$

Step 34.1.2

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

$5 \frac{1}{{(1 + 25 x^{2})}^{\frac{1}{2} + 1}}$

Step 34.2

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

$5 \frac{1}{{(1 + 25 x^{2})}^{\frac{1}{2} + \frac{2}{2}}}$

Step 34.3

Combine the numerators over the common denominator.

$5 \frac{1}{{(1 + 25 x^{2})}^{\frac{1 + 2}{2}}}$

Step 34.4

Add $1$ and $2$ .

$5 \frac{1}{{(1 + 25 x^{2})}^{\frac{3}{2}}}$

Step 35

Combine $5$ and $\frac{1}{{(1 + 25 x^{2})}^{\frac{3}{2}}}$ .

$\frac{5}{{(1 + 25 x^{2})}^{\frac{3}{2}}}$

Step 36

Reorder terms.

$\frac{5}{{(25 x^{2} + 1)}^{\frac{3}{2}}}$