Calculus Examples

cos (4 x)

$\cos (4 x)$

Step 1

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \cos (x)$ and

$g (x) = 4 x$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set

$u$ as

$4 x$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [4 x]$

Step 1.2

The derivative of

$\cos (u)$ with respect to

$u$ is

$- \sin (u)$ .

$- \sin (u) \frac{d}{d x} [4 x]$

Step 1.3

Replace all occurrences of

$u$ with

$4 x$ .

$- \sin (4 x) \frac{d}{d x} [4 x]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

Since

$4$ is constant with respect to

$x$ , the derivative of

$4 x$ with respect to

$x$ is

$4 \frac{d}{d x} [x]$ .

$- \sin (4 x) (4 \frac{d}{d x} [x])$

Step 2.2

Multiply

$4$ by

$- 1$ .

$- 4 \sin (4 x) \frac{d}{d x} [x]$

Step 2.3

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$- 4 \sin (4 x) \cdot 1$

Step 2.4

Multiply

$- 4$ by

$1$ .

$- 4 \sin (4 x)$