Derivative of x ln x: Formula, Steps and Examples

The derivative of x\ln x is \ln x + 1. It is a product, so the product rule does all the work.

Step by step with the product rule

The product rule says (fg)' = f'g + fg'. Take f = x and g = \ln x:

1. f' = 1.
2. g' = \frac{1}{x}, from the derivative of ln x.
3. Combine: 1\cdot\ln x + x\cdot\frac{1}{x} = \ln x + 1.

The x in front cancels the \frac{1}{x} from the log, leaving a clean answer.

Graph of x ln x and its derivative

Finding the minimum

Set the derivative to zero:

For x < \frac{1}{e} the derivative is negative, and for x > \frac{1}{e} it is positive, so this is a minimum. Its value is

The second derivative confirms it: \frac{d^2}{dx^2}x\ln x = \frac{1}{x}, which is positive for every x > 0, so the curve is concave up everywhere.

What happens near x = 0

The graph seems to start at the origin, even though x\ln x is not defined at 0. That is because \lim_{x \to 0^+} x\ln x = 0. Write it as \frac{\ln x}{1/x} and apply L'Hôpital's rule:

Meanwhile the slope \ln x + 1 goes to -\infty, so the curve leaves the origin almost vertically.

Related examples

Example 1. \frac{d}{dx}(x\ln x - x) = \ln x + 1 - 1 = \ln x. This shows x\ln x - x is an antiderivative of \ln x; see the integral of ln x.

Example 2. \frac{d}{dx}x^2\ln x = 2x\ln x + x^2\cdot\frac{1}{x} = 2x\ln x + x = x(2\ln x + 1).

Example 3. \frac{d}{dx}\frac{\ln x}{x} = \frac{\frac{1}{x}\cdot x - \ln x}{x^2} = \frac{1 - \ln x}{x^2}, by the quotient rule. Its maximum is at x = e.

Example 4. \frac{d}{dx}x\ln(3x) = \ln(3x) + 1, since \frac{d}{dx}\ln(3x) = \frac{1}{x} too.

Common mistakes

1. Multiplying the derivatives. 1\cdot\frac{1}{x} = \frac{1}{x} is not the product rule. You need both terms.
2. Forgetting the domain. x\ln x only exists for x > 0, so the minimum search only happens there.
3. Mixing it up with x^x. \frac{d}{dx}x^x = x^x(\ln x + 1). The factor \ln x + 1 appears because x^x = e^{x\ln x}; see the derivative of xˣ.

To go the other way, the integral of x ln x uses integration by parts. Practise more product rule problems on product and quotient rules.