AP Calculus AB and BC

Derivative of ln(x^3): The Log Rule Shortcut

Q: How is it different from ln(3x)?

\ln(3x) has derivative \frac{1}{x}, because a coefficient becomes an additive constant. An exponent becomes a multiplier instead.

Q: What about (ln x)^3?

Different function: its derivative is \frac{3(\ln x)^{2}}{x}.

The derivative of ln of x cubed is 3 over x. The logarithm rule lets you rewrite it as 3 ln x first, so the exponent becomes a constant multiplier and the derivative is three times the derivative of ln x.

\frac{d}{dx}\left[\ln\left(x^{3}\right)\right] = \frac{3}{x}

Simplify before differentiating

\ln\left(x^{3}\right) = 3\ln x \implies \frac{d}{dx} = \frac{3}{x}

The chain rule reaches the same answer with more work: $\frac{3x^{2}}{x^{3}} = \frac{3}{x}$ . Simplifying first is faster and less error prone.

Contrast with the coefficient case

An exponent comes out as a multiplier; a coefficient does not. That is why $\ln(3x)$ has derivative $\frac{1}{x}$ while $\ln\left(x^{3}\right)$ has derivative $\frac{3}{x}$ .

\frac{d}{dx}\ln(3x) = \frac{1}{x}, \qquad \frac{d}{dx}\ln\left(x^{3}\right) = \frac{3}{x}

Common mistakes

Answering $\frac{1}{x^{3}}$ , forgetting the chain rule entirely.
Answering $\frac{3x^{2}}{x^{3}}$ and leaving it unsimplified.
Confusing $\ln\left(x^{3}\right)$ with $\left(\ln x\right)^{3}$ , whose derivative is $\frac{3(\ln x)^{2}}{x}$ .

Check yourself, not just the answer

Type derivatives and get graded on mathematical equivalence, with rule-level hints when you miss, in the Derivative Practice Checker.

Frequently asked questions

What is the derivative of ln(x^3)?

It is $\frac{3}{x}$ .

How is it different from ln(3x)?

$\ln(3x)$ has derivative $\frac{1}{x}$ , because a coefficient becomes an additive constant. An exponent becomes a multiplier instead.

What about (ln x)^3?

Different function: its derivative is $\frac{3(\ln x)^{2}}{x}$ .