Derivative of ln(x²) vs (ln x)²: Formulas and Examples

The derivative of \ln(x^2) is \dfrac{2}{x}. Do not confuse it with (\ln x)^2, whose derivative is \dfrac{2\ln x}{x}.

Derivative of ln(x²): two ways

With the chain rule. The inside is u = x^2, so u' = 2x. Using "derivative of the inside over the inside" from the derivative of ln x:

With a log rule first. The power rule for logs brings the exponent down: \ln(x^2) = 2\ln\lvert x\rvert. Since \frac{d}{dx}\ln\lvert x\rvert = \frac{1}{x},

The absolute value matters because x^2 > 0 even when x < 0. So \ln(x^2) exists for negative x too, and the formula \frac{2}{x} is valid on both sides of 0.

Derivative of (ln x)²

Here the log is computed first and then squared. The outer function is a square, so use the chain rule with the power rule outside:

You will also see (\ln x)^2 written as \ln^2 x. It never equals \ln(x^2) in general. At x = e: \ln(e^2) = 2, but (\ln e)^2 = 1.

Graph of both functions

The graph shows the difference in sign. \ln(x^2) rises for x > 0 (its slope \frac{2}{x} is positive) and falls for x < 0. (\ln x)^2 has a minimum at x = 1, where its derivative \frac{2\ln x}{x} equals 0.

Side by side

More examples

Example 1. \frac{d}{dx}\ln(x^3) = \frac{3x^2}{x^3} = \frac{3}{x}. In general \frac{d}{dx}\ln(x^n) = \frac{n}{x}.

Example 2. \frac{d}{dx}\ln(x^2 + 1) = \frac{2x}{x^2 + 1}. Here nothing cancels, because the inside is not a pure power.

Example 3. \frac{d}{dx}(\ln x)^3 = \frac{3(\ln x)^2}{x}.

Example 4. \frac{d}{dx}\ln\!\left((x^2 + 1)^2\right) = 2\cdot\frac{2x}{x^2 + 1} = \frac{4x}{x^2 + 1}, using the log power rule first.

Common mistakes

1. Writing \frac{1}{x^2}. You must multiply by the inside derivative 2x.
2. Treating \ln^2 x as \ln(x^2). One squares the output, the other the input.
3. Writing \frac{2}{x} for (\ln x)^2. You dropped the \ln x that the power rule leaves behind.

For more chain rule practice with logs, see the derivative of ln(2x), logarithmic differentiation and derivatives of exponential and logarithmic functions.