Calculus Examples

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

Step 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 $f (x) = \tan (x)$ and $g (x) = \sec (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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Step 8.1.1

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

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

Step 8.1.2

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

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

Step 8.2

Add $1$ and $2$ .

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