How do you differentiate x^x?

To differentiate x^x, we first let y = x^x. (Note that x^x is not in the form x^c where c is a constant or a^x where a is a constant so the usual differentiation formulas cannot be used). The trick here is to take the natutral logarithm of both sides. Then you obtain, ln(y) = ln(x^x). From here you need to use the rule that ln(x^x) = xln(x). So currently we have ln(y) = xln(x). From here we can differentiate implicitly to get: 1/y multiplied by dy/dx = ln(x) + 1 (differentiate right hand side using product rule and left hand side using chain rule).The final step is to multiply through by y and substitute x^x back in for y. This gives you: dy/dx = x^x(ln(x) + 1).