Calculus Examples

$f (x) = 9 x^{9} - 5 \cos (x)$

Step 1

By the Sum Rule, the derivative of $9 x^{9} - 5 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [9 x^{9}] + \frac{d}{d x} [- 5 \cos (x)]$ .

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

Step 2

Evaluate $\frac{d}{d x} [9 x^{9}]$ .

Tap for more steps...

Step 2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{9}$ with respect to $x$ is $9 \frac{d}{d x} [x^{9}]$ .

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

Step 2.2

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

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

Step 2.3

Multiply $9$ by $9$ .

$81 x^{8} + \frac{d}{d x} [- 5 \cos (x)]$

Step 3

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

Tap for more steps...

Step 3.1

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

$81 x^{8} - 5 \frac{d}{d x} [\cos (x)]$

Step 3.2

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

$81 x^{8} - 5 (- \sin (x))$

Step 3.3

Multiply $- 1$ by $- 5$ .

$81 x^{8} + 5 \sin (x)$