Calculus Examples

$\cos (\arctan (2 x))$

Step 1

Simplify with factoring out.

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Step 1.1

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

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

Step 1.2

Factor $2$ out of $2 x$ .

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

Step 1.3

Simplify the expression.

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Step 1.3.1

Apply the product rule to $2 (x)$ .

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

Step 1.3.2

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

$\frac{d}{d x} [\frac{1}{\sqrt{1 + 4 x^{2}}}]$

Step 1.3.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + 4 x^{2}}$ as ${(1 + 4 x^{2})}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{1}{{(1 + 4 x^{2})}^{\frac{1}{2}}}]$

Step 1.3.4

Rewrite $\frac{1}{{(1 + 4 x^{2})}^{\frac{1}{2}}}$ as ${({(1 + 4 x^{2})}^{\frac{1}{2}})}^{- 1}$ .

$\frac{d}{d x} [{({(1 + 4 x^{2})}^{\frac{1}{2}})}^{- 1}]$

Step 1.3.5

Multiply the exponents in ${({(1 + 4 x^{2})}^{\frac{1}{2}})}^{- 1}$ .

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Step 1.3.5.1

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

$\frac{d}{d x} [{(1 + 4 x^{2})}^{\frac{1}{2} \cdot - 1}]$

Step 1.3.5.2

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

$\frac{d}{d x} [{(1 + 4 x^{2})}^{\frac{- 1}{2}}]$

Step 1.3.5.3

Move the negative in front of the fraction.

$\frac{d}{d x} [{(1 + 4 x^{2})}^{- \frac{1}{2}}]$

Step 2

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 $g (x) = 1 + 4 x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $1 + 4 x^{2}$ .

$\frac{d}{d u} [u^{- \frac{1}{2}}] \frac{d}{d x} [1 + 4 x^{2}]$

Step 2.2

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

$- \frac{1}{2} u^{- \frac{1}{2} - 1} \frac{d}{d x} [1 + 4 x^{2}]$

Step 2.3

Replace all occurrences of $u$ with $1 + 4 x^{2}$ .

$- \frac{1}{2} {(1 + 4 x^{2})}^{- \frac{1}{2} - 1} \frac{d}{d x} [1 + 4 x^{2}]$

Step 3

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

$- \frac{1}{2} {(1 + 4 x^{2})}^{- \frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [1 + 4 x^{2}]$

Step 4

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

$- \frac{1}{2} {(1 + 4 x^{2})}^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [1 + 4 x^{2}]$

Step 5

Combine the numerators over the common denominator.

$- \frac{1}{2} {(1 + 4 x^{2})}^{\frac{- 1 - 1 \cdot 2}{2}} \frac{d}{d x} [1 + 4 x^{2}]$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$- \frac{1}{2} {(1 + 4 x^{2})}^{\frac{- 1 - 2}{2}} \frac{d}{d x} [1 + 4 x^{2}]$

Step 6.2

Subtract $2$ from $- 1$ .

$- \frac{1}{2} {(1 + 4 x^{2})}^{\frac{- 3}{2}} \frac{d}{d x} [1 + 4 x^{2}]$

Step 7

Combine fractions.

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Step 7.1

Move the negative in front of the fraction.

$- \frac{1}{2} {(1 + 4 x^{2})}^{- \frac{3}{2}} \frac{d}{d x} [1 + 4 x^{2}]$

Step 7.2

Combine ${(1 + 4 x^{2})}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$- \frac{{(1 + 4 x^{2})}^{- \frac{3}{2}}}{2} \frac{d}{d x} [1 + 4 x^{2}]$

Step 7.3

Move ${(1 + 4 x^{2})}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [1 + 4 x^{2}]$

Step 8

By the Sum Rule, the derivative of $1 + 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [4 x^{2}]$ .

$- \frac{1}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}} (\frac{d}{d x} [1] + \frac{d}{d x} [4 x^{2}])$

Step 9

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

$- \frac{1}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}} (0 + \frac{d}{d x} [4 x^{2}])$

Step 10

Add $0$ and $\frac{d}{d x} [4 x^{2}]$ .

$- \frac{1}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [4 x^{2}]$

Step 11

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$- \frac{1}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}} (4 \frac{d}{d x} [x^{2}])$

Step 12

Simplify terms.

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Step 12.1

Multiply $4$ by $- 1$ .

$- 4 \frac{1}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [x^{2}]$

Step 12.2

Combine $- 4$ and $\frac{1}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}}$ .

$\frac{- 4}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [x^{2}]$

Step 12.3

Factor $2$ out of $- 4$ .

$\frac{2 \cdot - 2}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [x^{2}]$

Step 13

Cancel the common factors.

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Step 13.1

Factor $2$ out of $2 {(1 + 4 x^{2})}^{\frac{3}{2}}$ .

$\frac{2 \cdot - 2}{2 ({(1 + 4 x^{2})}^{\frac{3}{2}})} \frac{d}{d x} [x^{2}]$

Step 13.2

Cancel the common factor.

$\frac{2 \cdot - 2}{2 {(1 + 4 x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [x^{2}]$

Step 13.3

Rewrite the expression.

$\frac{- 2}{{(1 + 4 x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [x^{2}]$

Step 14

Move the negative in front of the fraction.

$- \frac{2}{{(1 + 4 x^{2})}^{\frac{3}{2}}} \frac{d}{d x} [x^{2}]$

Step 15

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

$- \frac{2}{{(1 + 4 x^{2})}^{\frac{3}{2}}} (2 x)$

Step 16

Combine fractions.

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Step 16.1

Multiply $2$ by $- 1$ .

$- 2 \frac{2}{{(1 + 4 x^{2})}^{\frac{3}{2}}} x$

Step 16.2

Combine $- 2$ and $\frac{2}{{(1 + 4 x^{2})}^{\frac{3}{2}}}$ .

$\frac{- 2 \cdot 2}{{(1 + 4 x^{2})}^{\frac{3}{2}}} x$

Step 16.3

Multiply $- 2$ by $2$ .

$\frac{- 4}{{(1 + 4 x^{2})}^{\frac{3}{2}}} x$

Step 16.4

Combine $\frac{- 4}{{(1 + 4 x^{2})}^{\frac{3}{2}}}$ and $x$ .

$\frac{- 4 x}{{(1 + 4 x^{2})}^{\frac{3}{2}}}$

Step 16.5

Move the negative in front of the fraction.

$- \frac{4 x}{{(1 + 4 x^{2})}^{\frac{3}{2}}}$