Calculus Examples

$4 \ln (4 x)$

Step 1

Since $4$ is constant with respect to $x$ , the derivative of $4 \ln (4 x)$ with respect to $x$ is $4 \frac{d}{d x} [\ln (4 x)]$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 4 x$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $4 x$ .

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

Step 2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

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

Step 2.3

Replace all occurrences of $u$ with $4 x$ .

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

Step 3

Differentiate.

Tap for more steps...

Step 3.1

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

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

Step 3.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

$\frac{4}{x}$