Calculus Examples

$arcsec (\frac{1}{x})$

Step 1

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

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Step 1.1

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

$\frac{d}{d u} [arcsec (u)] \frac{d}{d x} [\frac{1}{x}]$

Step 1.2

The derivative of $arcsec (u)$ with respect to $u$ is $\frac{1}{u \sqrt{u^{2} - 1}}$ .

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

Step 1.3

Replace all occurrences of $u$ with $\frac{1}{x}$ .

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

Step 2

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

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\sqrt{{(\frac{1}{x})}^{2} - 1}}{x}$ .

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

Step 4

Simplify the expression.

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Step 4.1

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

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

Step 4.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Multiply $x$ by $x^{- 2}$ by adding the exponents.

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Step 7.1

Multiply $x$ by $x^{- 2}$ .

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.2

Subtract $2$ from $1$ .

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Apply the product rule to $\frac{1}{x}$ .

$- \frac{1}{\sqrt{\frac{1^{2}}{x^{2}} - 1} x}$

Step 9.2

One to any power is one.

$- \frac{1}{\sqrt{\frac{1}{x^{2}} - 1} x}$