Calculus Examples

$f (x) = \ln (10 x)$

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 $g (x) = 10 x$ .

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To apply the Chain Rule, set $u$ as $10 x$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [10 x]$

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

$\frac{1}{u} \frac{d}{d x} [10 x]$

Replace all occurrences of $u$ with $10 x$ .

$\frac{1}{10 x} \frac{d}{d x} [10 x]$

Step 2

Differentiate.

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Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$\frac{1}{10 x} (10 \frac{d}{d x} [x])$

Simplify terms.

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Combine $10$ and $\frac{1}{10 x}$ .

$\frac{10}{10 x} \frac{d}{d x} [x]$

Cancel the common factor of $10$ .

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Cancel the common factor.

$\frac{10}{10 x} \frac{d}{d x} [x]$

Rewrite the expression.

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

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

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

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

$\frac{1}{x}$