Calculus Examples

$\cot (arcsec (\frac{u}{2}))$

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Rewrite $\frac{1}{{({(\frac{u}{2})}^{2} - 1)}^{\frac{1}{2}}}$ as ${({({(\frac{u}{2})}^{2} - 1)}^{\frac{1}{2}})}^{- 1}$ .

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Move the negative in front of the fraction.

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

Step 2

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

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Step 2.1

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

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

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $- 1$ .

$- \frac{1}{2} {({(\frac{u}{2})}^{2} - 1)}^{\frac{- 3}{2}} \frac{d}{d u} [{(\frac{u}{2})}^{2} - 1]$

Step 7

Differentiate using the Sum Rule.

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Step 7.1

Move the negative in front of the fraction.

$- \frac{1}{2} {({(\frac{u}{2})}^{2} - 1)}^{- \frac{3}{2}} \frac{d}{d u} [{(\frac{u}{2})}^{2} - 1]$

Step 7.2

Combine fractions.

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Step 7.2.1

Combine ${({(\frac{u}{2})}^{2} - 1)}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$- \frac{{({(\frac{u}{2})}^{2} - 1)}^{- \frac{3}{2}}}{2} \frac{d}{d u} [{(\frac{u}{2})}^{2} - 1]$

Step 7.2.2

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

$- \frac{1}{2 {({(\frac{u}{2})}^{2} - 1)}^{\frac{3}{2}}} \frac{d}{d u} [{(\frac{u}{2})}^{2} - 1]$

Step 7.3

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Replace all occurrences of $u_{2}$ with $\frac{u}{2}$ .

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

Step 9

Differentiate.

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Step 9.1

Combine $2$ and $\frac{u}{2}$ .

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

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Cancel the common factor.

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

Step 9.2.2

Divide $u$ by $1$ .

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

Multiply $\frac{u}{2}$ by $1$ .

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

Step 9.7

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

$- \frac{1}{2 {({(\frac{u}{2})}^{2} - 1)}^{\frac{3}{2}}} (\frac{u}{2} + 0)$

Step 9.8

Combine fractions.

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Step 9.8.1

Add $\frac{u}{2}$ and $0$ .

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

Step 9.8.2

Multiply $\frac{u}{2}$ by $\frac{1}{2 {({(\frac{u}{2})}^{2} - 1)}^{\frac{3}{2}}}$ .

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

Step 9.8.3

Multiply $2$ by $2$ .

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

Step 10

Simplify.

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Step 10.1

Apply the product rule to $\frac{u}{2}$ .

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

Step 10.2

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

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