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The derivative of ln

The derivative of the natural logarithm of x is 1 divided by x, for positive x. If a function sits inside the logarithm, you divide the derivative of the inside by the inside itself. The denominator is therefore the inner function, not its derivative. Ln increases for every positive x, so the derivative is positive there. You can only differentiate ln where the argument is positive.

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

$(\ln x)'$ at $x = 4$ is $\dfrac{1}{4} = 0.25$ .

$(\ln(x^2 + 1))'$ at $x = 2$ is $\dfrac{2 \cdot 2}{2^2 + 1} = 0.8$ .

The fraction is the derivative of the inner function divided by the inner function.

Practise differentiation rules in practice for free →

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Part of Mathematics S2: Differentiation rules in practice.