Calculus Examples

ln (4 x)

$\ln (4 x)$

Step 1

Differentiate using the chain rule, which states that

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

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

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

$g (x) = 4 x$ .

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Step 1.1

To apply the Chain Rule, set

$u$ as

$4 x$ .

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

Step 1.2

The derivative of

$\ln (u)$ with respect to

$u$ is

$\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [4 x]$

Step 1.3

Replace all occurrences of

$u$ with

$4 x$ .

$\frac{1}{4 x} \frac{d}{d x} [4 x]$

Step 2

Differentiate.

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Step 2.1

Since

$4$ is constant with respect to

$x$ , the derivative of

$4 x$ with respect to

$x$ is

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

$\frac{1}{4 x} (4 \frac{d}{d x} [x])$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine

$4$ and

$\frac{1}{4 x}$ .

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

Step 2.2.2

Cancel the common factor of

$4$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

$\frac{1}{x} \frac{d}{d x} [x]$

Step 2.3

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$\frac{1}{x} \cdot 1$

Step 2.4

Multiply

$\frac{1}{x}$ by

$1$ .

$\frac{1}{x}$