Calculus Examples

$\log_{5} (\ln (x))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{5} (x)$ and $g (x) = \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\ln (x)$ .

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

Step 1.2

The derivative of $\log_{5} (u)$ with respect to $u$ is $\frac{1}{u \ln (5)}$ .

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

Step 1.3

Replace all occurrences of $u$ with $\ln (x)$ .

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

Step 2

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

$\frac{1}{\ln (x) \ln (5)} \cdot \frac{1}{x}$

Step 3

Combine fractions.

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Step 3.1

Multiply $\frac{1}{\ln (x) \ln (5)}$ by $\frac{1}{x}$ .

$\frac{1}{\ln (x) \ln (5) x}$

Step 3.2

Reorder terms.

$\frac{1}{x \ln (5) \ln (x)}$