Calculus Examples

\sin (4 x)

$\sin (4 x)$

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)

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

f (x) = \sin (x)

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

g (x) = 4 x

$g (x) = 4 x$ .

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Step 1.1

To apply the Chain Rule, set

u

$u$ as

4 x

$4 x$ .

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

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

Step 1.2

The derivative of

\sin (u)

$\sin (u)$ with respect to

u

$u$ is

\cos (u)

$\cos (u)$ .

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

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

Step 1.3

Replace all occurrences of

u

$u$ with

4 x

$4 x$ .

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

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

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

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

Step 2

Differentiate.

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Step 2.1

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

4 x

$4 x$ with respect to

x

$x$ is

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

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

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

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

Step 2.2

Differentiate using the Power Rule which states that

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

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

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

\cos (4 x) (4 \cdot 1)

$\cos (4 x) (4 \cdot 1)$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply

4

$4$ by

1

$1$ .

\cos (4 x) \cdot 4

$\cos (4 x) \cdot 4$

Step 2.3.2

Move

4

$4$ to the left of

\cos (4 x)

$\cos (4 x)$ .

4 \cos (4 x)

$4 \cos (4 x)$

4 \cos (4 x)

$4 \cos (4 x)$

4 \cos (4 x)

$4 \cos (4 x)$