Calculus Examples

$\log_{a} (x)$

Step 1

Rewrite $\log_{a} (x)$ using the change of base formula.

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Step 1.1

The change of base rule can be used if $a$ and $b$ are greater than 0 and not equal to 1, and $x$ is greater than 0.

$\frac{d}{d x} [\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}]$

Step 1.2

Substitute in values for the variables in the change of base formula, using $b = e$ .

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

Step 2

Since $\frac{1}{\ln (a)}$ is constant with respect to $x$ , the derivative of $\frac{\ln (x)}{\ln (a)}$ with respect to $x$ is $\frac{1}{\ln (a)} \frac{d}{d x} [\ln (x)]$ .

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

Reorder terms.

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