Calculus Examples

$3 \ln (5 x)$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 \ln (5 x)$ with respect to $x$ is $3 \frac{d}{d x} [\ln (5 x)]$ .

$3 \frac{d}{d x} [\ln (5 x)]$

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 $g (x) = 5 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$3 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [5 x])$

Step 2.2

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

$3 (\frac{1}{u} \frac{d}{d x} [5 x])$

Step 2.3

Replace all occurrences of $u$ with $5 x$ .

$3 (\frac{1}{5 x} \frac{d}{d x} [5 x])$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{5 x}$ and $3$ .

$\frac{3}{5 x} \frac{d}{d x} [5 x]$

Step 3.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{3}{5 x} (5 \frac{d}{d x} [x])$

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $5$ and $\frac{3}{5 x}$ .

$\frac{5 \cdot 3}{5 x} \frac{d}{d x} [x]$

Step 3.3.2

Multiply $5$ by $3$ .

$\frac{15}{5 x} \frac{d}{d x} [x]$

Step 3.3.3

Cancel the common factor of $15$ and $5$ .

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Step 3.3.3.1

Factor $5$ out of $15$ .

$\frac{5 \cdot 3}{5 x} \frac{d}{d x} [x]$

Step 3.3.3.2

Cancel the common factors.

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Step 3.3.3.2.1

Factor $5$ out of $5 x$ .

$\frac{5 \cdot 3}{5 (x)} \frac{d}{d x} [x]$

Step 3.3.3.2.2

Cancel the common factor.

$\frac{5 \cdot 3}{5 x} \frac{d}{d x} [x]$

Step 3.3.3.2.3

Rewrite the expression.

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

Step 3.4

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

$\frac{3}{x} \cdot 1$

Step 3.5

Multiply $\frac{3}{x}$ by $1$ .

$\frac{3}{x}$