Calculus Examples

$(\sec (x))$

Step 1

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

$f' (x) = \sec (x) \tan (x)$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sec (x)$ and $g (x) = \tan (x)$ .

$\sec (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\sec (x)]$

Step 2.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.2

Add $1$ and $2$ .

$\sec^{3} (x) + \tan (x) \frac{d}{d x} [\sec (x)]$

Step 2.4

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

$\sec^{3} (x) + \tan (x) (\sec (x) \tan (x))$

Step 2.5

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

$\sec^{3} (x) + \tan^{1} (x) \tan (x) \sec (x)$

Step 2.6

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

$\sec^{3} (x) + \tan^{1} (x) \tan^{1} (x) \sec (x)$

Step 2.7

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

$\sec^{3} (x) + {\tan (x)}^{1 + 1} \sec (x)$

Step 2.8

Add $1$ and $1$ .

$\sec^{3} (x) + \tan^{2} (x) \sec (x)$

Step 2.9

Reorder terms.

$f'' (x) = \tan^{2} (x) \sec (x) + \sec^{3} (x)$