Derivative of ln x

Q: What is the derivative of ln x?

\frac{d}{dx}\left[\ln\left(x\right)\right] = \frac{1}{x}.

\frac{d}{dx}\left[\ln\left(x\right)\right] = \frac{1}{x}

The natural log grows ever more slowly, and its slope at x is exactly 1/x: at x = 1 the slope is 1, at x = 10 only 0.1. The derivative only exists for x > 0, where ln x is defined.

Derivation

Logarithm rule
$\frac{d}{dx}\left[\ln\left(x\right)\right] = \frac{1}{x}$

f(x) = ln x
f′(x)

Drag the graph · at x = 3.92 the slope is 0.2551, the height of the dashed curve.

Open in the derivative calculator

Questions students ask

What is the derivative of ln x?

$\frac{d}{dx}\left[\ln\left(x\right)\right] = \frac{1}{x}$ .

How do you find the derivative of ln x?

The natural log grows ever more slowly, and its slope at x is exactly 1/x: at x = 1 the slope is 1, at x = 10 only 0.1. The derivative only exists for x > 0, where ln x is defined.

What is the integral of ln x?

See the integral of ln x page for the antiderivative with steps.