Calculus Examples

$\sin^{7} (3 x)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d ?} [f (g (?))]$ is $f' (g (?)) g' (?)$ where $f (?) = ?^{7}$ and $g (?) = \sin (3 x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\sin (3 x)$ .

$\frac{d}{d u} [u^{7}] \frac{d}{d ?} [\sin (3 x)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

$7 u^{6} \frac{d}{d ?} [\sin (3 x)]$

Step 1.3

Replace all occurrences of $u$ with $\sin (3 x)$ .

$7 \sin^{6} (3 x) \frac{d}{d ?} [\sin (3 x)]$

Step 2

Since $\sin (3 x)$ is constant with respect to $?$ , the derivative of $\sin (3 x)$ with respect to $?$ is $0$ .

$7 \sin^{6} (3 x) \cdot 0$

Step 3

Combine terms.

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Step 3.1

Multiply $0$ by $7$ .

$0 \sin^{6} (3 x)$

Step 3.2

Multiply $0$ by $\sin^{6} (3 x)$ .

$0$