Calculus Examples

$(\frac{1}{3}) \cdot \cos (8 x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \frac{1}{3}$ and $g (x) = \cos (8 x)$ .

$\frac{1}{3} \frac{d}{d x} [\cos (8 x)] + \cos (8 x) \frac{d}{d x} [\frac{1}{3}]$

Step 2

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 $g (x) = 8 x$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $8 x$ .

$\frac{1}{3} (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [8 x]) + \cos (8 x) \frac{d}{d x} [\frac{1}{3}]$

Step 2.2

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

$\frac{1}{3} (- \sin (u) \frac{d}{d x} [8 x]) + \cos (8 x) \frac{d}{d x} [\frac{1}{3}]$

Step 2.3

Replace all occurrences of $u$ with $8 x$ .

$\frac{1}{3} (- \sin (8 x) \frac{d}{d x} [8 x]) + \cos (8 x) \frac{d}{d x} [\frac{1}{3}]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$\frac{1}{3} (- \sin (8 x) (8 \frac{d}{d x} [x])) + \cos (8 x) \frac{d}{d x} [\frac{1}{3}]$

Step 3.2

Multiply $8$ by $- 1$ .

$\frac{1}{3} (- 8 \sin (8 x) \frac{d}{d x} [x]) + \cos (8 x) \frac{d}{d x} [\frac{1}{3}]$

Step 3.3

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

$\frac{1}{3} (- 8 \sin (8 x) \cdot 1) + \cos (8 x) \frac{d}{d x} [\frac{1}{3}]$

Step 3.4

Multiply $- 8$ by $1$ .

$\frac{1}{3} (- 8 \sin (8 x)) + \cos (8 x) \frac{d}{d x} [\frac{1}{3}]$

Step 3.5

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3}$ with respect to $x$ is $0$ .

$\frac{1}{3} (- 8 \sin (8 x)) + \cos (8 x) \cdot 0$

Step 4

Combine terms.

Tap for more steps...

Step 4.1

Combine $- 8$ and $\frac{1}{3}$ .

$\frac{- 8}{3} \sin (8 x) + \cos (8 x) \cdot 0$

Step 4.2

Combine $\frac{- 8}{3}$ and $\sin (8 x)$ .

$\frac{- 8 \sin (8 x)}{3} + \cos (8 x) \cdot 0$

Step 4.3

Move the negative in front of the fraction.

$- \frac{8 \sin (8 x)}{3} + \cos (8 x) \cdot 0$

Step 4.4

Multiply $\cos (8 x)$ by $0$ .

$- \frac{8 \sin (8 x)}{3} + 0$

Step 4.5

Add $- \frac{8 \sin (8 x)}{3}$ and $0$ .

$- \frac{8 \sin (8 x)}{3}$