Calculus Examples

$y = {(\sec (x) + \tan (x))}^{15}$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{15}$ and $g (x) = \sec (x) + \tan (x)$ .

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

$\frac{d}{d u} [u^{15}] \frac{d}{d x} [\sec (x) + \tan (x)]$

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 15$ .

$15 u^{14} \frac{d}{d x} [\sec (x) + \tan (x)]$

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

$15 {(\sec (x) + \tan (x))}^{14} \frac{d}{d x} [\sec (x) + \tan (x)]$

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

$15 {(\sec (x) + \tan (x))}^{14} (\frac{d}{d x} [\sec (x)] + \frac{d}{d x} [\tan (x)])$

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

$15 {(\sec (x) + \tan (x))}^{14} (\sec (x) \tan (x) + \frac{d}{d x} [\tan (x)])$

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

$15 {(\sec (x) + \tan (x))}^{14} (\sec (x) \tan (x) + \sec^{2} (x))$