Calculus Examples

$\cos (6 \sin (x))$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $6 \sin (x)$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [6 \sin (x)]$

Step 1.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$- \sin (u) \frac{d}{d x} [6 \sin (x)]$

Step 1.3

Replace all occurrences of $u$ with $6 \sin (x)$ .

$- \sin (6 \sin (x)) \frac{d}{d x} [6 \sin (x)]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \sin (x)$ with respect to $x$ is $6 \frac{d}{d x} [\sin (x)]$ .

$- \sin (6 \sin (x)) (6 \frac{d}{d x} [\sin (x)])$

Step 2.2

Multiply $6$ by $- 1$ .

$- 6 \sin (6 \sin (x)) \frac{d}{d x} [\sin (x)]$

Step 3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$- 6 \sin (6 \sin (x)) \cos (x)$

Step 4

Reorder the factors of $- 6 \sin (6 \sin (x)) \cos (x)$ .

$- 6 \cos (x) \sin (6 \sin (x))$