Calculus Examples

$g (x) = \cot^{2} (2 \sqrt{x}) + \sqrt{\tan (x) - 1}$

Step 1

By the Sum Rule, the derivative of $\cot^{2} (2 \sqrt{x}) + \sqrt{\tan (x) - 1}$ with respect to $x$ is $\frac{d}{d x} [\cot^{2} (2 \sqrt{x})] + \frac{d}{d x} [\sqrt{\tan (x) - 1}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\cot^{2} (2 \sqrt{x})]$ .

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Step 2.1

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

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cot (2 x^{\frac{1}{2}})$ .

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $\cot (2 x^{\frac{1}{2}})$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u_{1}$ with $\cot (2 x^{\frac{1}{2}})$ .

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cot (x)$ and $g (x) = 2 x^{\frac{1}{2}}$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{2}$ as $2 x^{\frac{1}{2}}$ .

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

Step 2.3.2

The derivative of $\cot (u_{2})$ with respect to $u_{2}$ is $- \csc^{2} (u_{2})$ .

$2 \cot (2 x^{\frac{1}{2}}) (- \csc^{2} (u_{2}) \frac{d}{d x} [2 x^{\frac{1}{2}}]) + \frac{d}{d x} [\sqrt{\tan (x) - 1}]$

Step 2.3.3

Replace all occurrences of $u_{2}$ with $2 x^{\frac{1}{2}}$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Multiply $- 1$ by $2$ .

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

Step 2.9.2

Subtract $2$ from $1$ .

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

Step 2.10

Move the negative in front of the fraction.

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

Step 2.11

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

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Cancel the common factor.

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

Step 2.15

Rewrite the expression.

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

Step 2.16

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

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

Step 2.17

Multiply $- 1$ by $2$ .

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

Step 2.18

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

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

Step 2.19

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

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

Step 2.20

Move $- 2$ to the left of $\csc^{2} (2 x^{\frac{1}{2}})$ .

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

Step 2.21

Move the negative in front of the fraction.

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

Step 3

Evaluate $\frac{d}{d x} [\sqrt{\tan (x) - 1}]$ .

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Step 3.1

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

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

Step 3.2

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 $g (x) = \tan (x) - 1$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{3}$ as $\tan (x) - 1$ .

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{d}{d u_{3}} [{u_{3}}^{\frac{1}{2}}] \frac{d}{d x} [\tan (x) - 1]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = \frac{1}{2}$ .

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{1}{2} {u_{3}}^{\frac{1}{2} - 1} \frac{d}{d x} [\tan (x) - 1]$

Step 3.2.3

Replace all occurrences of $u_{3}$ with $\tan (x) - 1$ .

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

Step 3.3

By the Sum Rule, the derivative of $\tan (x) - 1$ with respect to $x$ is $\frac{d}{d x} [\tan (x)] + \frac{d}{d x} [- 1]$ .

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

Step 3.4

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

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

Step 3.5

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

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{1}{2} {(\tan (x) - 1)}^{\frac{1}{2} - 1} (\sec^{2} (x) + 0)$

Step 3.6

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

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{1}{2} {(\tan (x) - 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (\sec^{2} (x) + 0)$

Step 3.7

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

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{1}{2} {(\tan (x) - 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (\sec^{2} (x) + 0)$

Step 3.8

Combine the numerators over the common denominator.

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{1}{2} {(\tan (x) - 1)}^{\frac{1 - 1 \cdot 2}{2}} (\sec^{2} (x) + 0)$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $- 1$ by $2$ .

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{1}{2} {(\tan (x) - 1)}^{\frac{1 - 2}{2}} (\sec^{2} (x) + 0)$

Step 3.9.2

Subtract $2$ from $1$ .

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{1}{2} {(\tan (x) - 1)}^{\frac{- 1}{2}} (\sec^{2} (x) + 0)$

Step 3.10

Move the negative in front of the fraction.

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{1}{2} {(\tan (x) - 1)}^{- \frac{1}{2}} (\sec^{2} (x) + 0)$

Step 3.11

Add $\sec^{2} (x)$ and $0$ .

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{1}{2} {(\tan (x) - 1)}^{- \frac{1}{2}} \sec^{2} (x)$

Step 3.12

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

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{{(\tan (x) - 1)}^{- \frac{1}{2}}}{2} \sec^{2} (x)$

Step 3.13

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

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{{(\tan (x) - 1)}^{- \frac{1}{2}} \sec^{2} (x)}{2}$

Step 3.14

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

$- \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}} + \frac{\sec^{2} (x)}{2 {(\tan (x) - 1)}^{\frac{1}{2}}}$

Step 4

Reorder terms.

$\frac{\sec^{2} (x)}{2 {(\tan (x) - 1)}^{\frac{1}{2}}} - \frac{2 \csc^{2} (2 x^{\frac{1}{2}}) \cot (2 x^{\frac{1}{2}})}{x^{\frac{1}{2}}}$