Differentiate x^x with logarithmic differentiation
This short walkthrough shows the standard logarithmic differentiation method for x^x. It keeps the algebra compact, shows where the chain rule and product rule appear, and ends with the final derivative for real x > 0.
95-second vertical explainer on logarithmic differentiation for x^x.
What this covers
Set y = x^x and take natural logs so the exponent comes down.
Differentiate ln y = x ln x using chain rule on the left and product rule on the right.
Substitute y = x^x back in to get x^x(ln x + 1).
For real-valued work, assume x > 0 so that ln x is defined.