Calculus Examples

$f (x) = \frac{1}{\sec (\frac{1}{x})}$

Step 1

Rewrite $\sec (\frac{1}{x})$ in terms of sines and cosines.

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

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (\frac{1}{x})}$ .

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

Step 3

Multiply $\cos (\frac{1}{x})$ by $1$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

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

Step 4.3

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

$- \sin (\frac{1}{x}) \frac{d}{d x} [\frac{1}{x}]$

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Multiply.

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Step 5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.2

Multiply $\sin (\frac{1}{x})$ by $1$ .

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

Step 6

Simplify.

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Step 6.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sin (\frac{1}{x}) \frac{1}{x^{2}}$

Step 6.2

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

$\frac{\sin (\frac{1}{x})}{x^{2}}$