Skill: Differentiate logarithmic functions

For logarithmic functions, use these derivative rules:

$$\frac{d}{dx}\ln x=\frac{1}{x}$$

and

$$\frac{d}{dx}\log_a x=\frac{1}{x\ln a},$$

where $a>0$ and $a\ne 1$.

If the logarithm contains a function $g(x)$, use the chain rule:

$$\frac{d}{dx}\ln(g(x))=\frac{g'(x)}{g(x)}$$

and

$$\frac{d}{dx}\log_a(g(x)) =\frac{g'(x)}{g(x)\ln a}.$$

The inside of every logarithm must be positive.

Example: Differentiate

$$f(x)=\ln(3x^2+1)+\log_2(5x-1).$$

First, identify the inside functions:

• For $\ln(3x^2+1)$, let $g(x)=3x^2+1$, so $g'(x)=6x$.
• For $\log_2(5x-1)$, let $h(x)=5x-1$, so $h'(x)=5$.

Apply the chain rule to the first term:

$$\frac{d}{dx}\ln(3x^2+1) =\frac{6x}{3x^2+1}.$$

Apply the logarithm rule to the second term. Since the base is $2$, include the factor $\ln 2$:

$$\frac{d}{dx}\log_2(5x-1) =\frac{5}{(5x-1)\ln 2}.$$

Therefore,

$$\boxed{f'(x)=\frac{6x}{3x^2+1} +\frac{5}{(5x-1)\ln 2}}.$$

The domain requires $5x-1>0$, so this derivative applies for $x>\frac15$.