Medium

The Derivative of the Natural Logarithm

Simple Explanation

The derivative of the natural logarithm, ln x, is simply the reciprocal of x: d/dx[ln x] = 1/x.

Why Do We Need It?

This surprisingly simple result is essential for differentiating any expression involving natural logarithms, and connects logarithms directly to the power rule's "missing case" (x⁻¹, whose antiderivative is ln x).

See It

The tangent line to y=ln(x) at (1,0), with slope 1

Formula

The Derivative of the Natural Logarithm

d/dx[ln x] = 1/x

The derivative of the natural logarithm is simply the reciprocal of x.

x: — a positive real number (the domain of ln x)

When to use it: Whenever differentiating an expression involving ln x.

Worked Example

Differentiate an expression with the natural logarithm

Find the derivative of f(x)=5ln(x), and evaluate f'(2).

Why Does This Work?

Since eˣ and ln x are inverse functions, and eˣ is its own derivative, applying the rule for differentiating an inverse function to ln x produces exactly 1/x — a clean result that also matches the tangent line shown, which has slope 1 at x=1 (since 1/1=1).

Real-Life Example

Finding the rate of change of perceived sound loudness

Perceived loudness (in decibels) is related to actual sound intensity through a logarithmic (natural log-based) formula.

The derivative of the natural logarithm gives the rate at which perceived loudness changes as actual sound intensity changes.

Practice

Find f'(4) for f(x)=ln(x).

Medium

Common mistake

Confusing the derivative of ln(x) (a simple reciprocal, 1/x) with power-rule-style derivatives — ln(x)'s derivative does not follow the "bring down the exponent" pattern at all.

Quick Review

d/dx[ln x] = 1/x.
A remarkably simple result, given how complex ln x itself looks.
Connects to the "missing case" of the power rule (the antiderivative of x⁻¹).