Calculus Examples

$\tan^{6} (x)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = \tan (x)$ .

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Step 1.1

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

$\frac{d}{d u} [u^{6}] \frac{d}{d x} [\tan (x)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d x} [\tan (x)]$

Step 1.3

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

$6 \tan^{5} (x) \frac{d}{d x} [\tan (x)]$

Step 2

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

$6 \tan^{5} (x) \sec^{2} (x)$