Calculus Examples

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

Step 1

Rewrite $1$ as $1^{2}$ .

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

Step 2

Rewrite $\frac{1^{2}}{\sec^{2} (x)}$ as ${(\frac{1}{\sec (x)})}^{2}$ .

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

Step 3

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 4

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

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

Step 5

Multiply $\cos (x)$ by $1$ .

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

Step 6

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 $g (x) = \cos (x)$ .

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

$2 \cos (x) (- \sin (x))$

Step 8

Multiply $- 1$ by $2$ .

$- 2 \cos (x) \sin (x)$