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The derivative of ln

The derivative of $\ln x$ is $1/x$ for positive $x$ . If a function sits inside the logarithm, you divide the derivative of the inside by the inside itself. That is the chain rule for ln. The denominator is therefore the original inner function, not its derivative. You can check with a number: the slope should be positive where the function increases, and ln increases for every positive $x$ .

Try it yourself

The derivative as a slope

(\ln x)' = \frac{1}{x}

the basic case

(\ln u)' = \frac{u'}{u}

with an inner function

Symbols

$u$	the inner function, positive
$u'$	the derivative of the inner function

Example

$(\ln(4x))' = \dfrac{1}{4x}\cdot 4 = \dfrac{1}{x}$ , which is $0.25$ when $x = 4$ . $(\ln(x^2 + 1))'$ at $x = 2$ is $\dfrac{4}{5} = 0.8$ .

Divide by the inner function, and multiply by its derivative. The order in the fraction is

u'/u

Practise differentiation rules in practice for free →

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Part of Mathematics R1: Differentiation rules in practice.