Calculus Examples

$f (x) = \frac{1}{\sqrt{\cos (x)}}$

Step 1

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

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

Step 2

Convert from $\frac{1}{{\cos (x)}^{\frac{1}{2}}}$ to ${\sec (x)}^{\frac{1}{2}}$ .

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

Step 3

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

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To apply the Chain Rule, set $u$ as $\sec (x)$ .

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

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

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

Replace all occurrences of $u$ with $\sec (x)$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Multiply $- 1$ by $2$ .

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

Subtract $2$ from $1$ .

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

Step 8

Combine fractions.

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Move the negative in front of the fraction.

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

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

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

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

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

Step 9

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

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

Step 10

Combine fractions.

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

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

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

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

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

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

Step 11

Multiply $\sec (x)$ by ${\sec (x)}^{- \frac{1}{2}}$ by adding the exponents.

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Move ${\sec (x)}^{- \frac{1}{2}}$ .

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

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

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Raise $\sec (x)$ to the power of $1$ .

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $2$ .

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