What is the derivative of x to the x?
The base and exponent both vary, so neither the ordinary power rule nor the simple exponential rule works alone.
LaTeX article Updated July 11, 2026
Why the usual rules do not fit
The power rule assumes a constant exponent. The derivative of a to the x assumes a constant base. In x to the x, both move, so applying either rule by itself drops part of the change.
Logarithmic differentiation turns the exponent into multiplication, where the product rule can see both contributions.
The logarithmic step
Set y equal to x to the x and take natural logs: ln y = x ln x. Differentiate implicitly, remembering that y is a function of x.
Multiplying by y at the end restores the original exponential expression.
Domain note
The real-valued logarithmic derivation is cleanest for x greater than zero. Extensions to selected negative rational inputs require more careful domain analysis and do not define one smooth real function on all negative x.
Worked example
Common mistakes
- Writing x·x^(x−1) and treating the exponent as constant.
- Forgetting the chain-rule factor y′/y after differentiating ln y.
- Omitting the positive-domain assumption.
Keep these ideas
- Variable exponents suggest logarithms.
- Differentiate x ln x with the product rule.
- Substitute y = xˣ back only at the end.