Calculus Examples

\tan^{3} (x)

$\tan^{3} (x)$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

f' (g (x)) g' (x)

$f' (g (x)) g' (x)$ where

f (x) = x^{3}

$f (x) = x^{3}$ and

g (x) = \tan (x)

$g (x) = \tan (x)$ .

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Step 1.1

To apply the Chain Rule, set

u

$u$ as

\tan (x)

$\tan (x)$ .

\frac{d}{d u} [u^{3}] \frac{d}{d x} [\tan (x)]

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

Step 1.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = 3

$n = 3$ .

3 u^{2} \frac{d}{d x} [\tan (x)]

$3 u^{2} \frac{d}{d x} [\tan (x)]$

Step 1.3

Replace all occurrences of

u

$u$ with

\tan (x)

$\tan (x)$ .

3 \tan^{2} (x) \frac{d}{d x} [\tan (x)]

$3 \tan^{2} (x) \frac{d}{d x} [\tan (x)]$

3 \tan^{2} (x) \frac{d}{d x} [\tan (x)]

$3 \tan^{2} (x) \frac{d}{d x} [\tan (x)]$

Step 2

The derivative of

\tan (x)

$\tan (x)$ with respect to

x

$x$ is

\sec^{2} (x)

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

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

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

Step 3

Reorder the factors of

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

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

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

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