How do you differentiate x^x?
To differentiate xx, we first let y = xx. (Note that xx is not in the form xc where c is a constant or ax 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(xx). From here you need to use the rule that ln(xx) = 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 xx back in for y. This gives you: dy/dx = xx(ln(x) + 1).
