Calculus Examples

${(\sec (x) + \tan (x))}^{5}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = \sec (x) + \tan (x)$ .

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Step 1.1

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

$\frac{d}{d u} [u^{5}] \frac{d}{d x} [\sec (x) + \tan (x)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$5 u^{4} \frac{d}{d x} [\sec (x) + \tan (x)]$

Step 1.3

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

$5 {(\sec (x) + \tan (x))}^{4} \frac{d}{d x} [\sec (x) + \tan (x)]$

Step 2

By the Sum Rule, the derivative of $\sec (x) + \tan (x)$ with respect to $x$ is $\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)]$ .

$5 {(\sec (x) + \tan (x))}^{4} (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)])$

Step 3

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

$5 {(\sec (x) + \tan (x))}^{4} (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)])$

Step 4

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

$5 {(\sec (x) + \tan (x))}^{4} (\sec (x) \tan (x) + \sec^{2} (x))$