Calculus Examples

$y = {(\sin (x) + 5 \cos (x))}^{9}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{9}$ and $g (x) = \sin (x) + 5 \cos (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\sin (x) + 5 \cos (x)$ .

$\frac{d}{d u} [u^{9}] \frac{d}{d x} [\sin (x) + 5 \cos (x)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 9$ .

$9 u^{8} \frac{d}{d x} [\sin (x) + 5 \cos (x)]$

Step 1.3

Replace all occurrences of $u$ with $\sin (x) + 5 \cos (x)$ .

$9 {(\sin (x) + 5 \cos (x))}^{8} \frac{d}{d x} [\sin (x) + 5 \cos (x)]$

Step 2

By the Sum Rule, the derivative of $\sin (x) + 5 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [5 \cos (x)]$ .

$9 {(\sin (x) + 5 \cos (x))}^{8} (\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [5 \cos (x)])$

Step 3

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

$9 {(\sin (x) + 5 \cos (x))}^{8} (\cos (x) + \frac{d}{d x} [5 \cos (x)])$

Step 4

Evaluate $\frac{d}{d x} [5 \cos (x)]$ .

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Step 4.1

Since $5$ is constant with respect to $x$ , the derivative of $5 \cos (x)$ with respect to $x$ is $5 \frac{d}{d x} [\cos (x)]$ .

$9 {(\sin (x) + 5 \cos (x))}^{8} (\cos (x) + 5 \frac{d}{d x} [\cos (x)])$

Step 4.2

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

$9 {(\sin (x) + 5 \cos (x))}^{8} (\cos (x) + 5 (- \sin (x)))$

Step 4.3

Multiply $- 1$ by $5$ .

$9 {(\sin (x) + 5 \cos (x))}^{8} (\cos (x) - 5 \sin (x))$