Calculus Examples

$f (x) = 2 \ln (3 x)$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 \ln (3 x)$ with respect to $x$ is $2 \frac{d}{d x} [\ln (3 x)]$ .

$2 \frac{d}{d x} [\ln (3 x)]$

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 $g (x) = 3 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x$ .

$2 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x])$

Step 2.2

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

$2 (\frac{1}{u} \frac{d}{d x} [3 x])$

Step 2.3

Replace all occurrences of $u$ with $3 x$ .

$2 (\frac{1}{3 x} \frac{d}{d x} [3 x])$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{3 x}$ and $2$ .

$\frac{2}{3 x} \frac{d}{d x} [3 x]$

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{2}{3 x} (3 \frac{d}{d x} [x])$

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $3$ and $\frac{2}{3 x}$ .

$\frac{3 \cdot 2}{3 x} \frac{d}{d x} [x]$

Step 3.3.2

Multiply $3$ by $2$ .

$\frac{6}{3 x} \frac{d}{d x} [x]$

Step 3.3.3

Cancel the common factor of $6$ and $3$ .

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Step 3.3.3.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 2}{3 x} \frac{d}{d x} [x]$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $3$ out of $3 x$ .

$\frac{3 \cdot 2}{3 (x)} \frac{d}{d x} [x]$

Step 3.3.3.2.2

Cancel the common factor.

$\frac{3 \cdot 2}{3 x} \frac{d}{d x} [x]$

Step 3.3.3.2.3

Rewrite the expression.

$\frac{2}{x} \frac{d}{d x} [x]$

Step 3.4

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

$\frac{2}{x} \cdot 1$

Step 3.5

Multiply $\frac{2}{x}$ by $1$ .

$\frac{2}{x}$