Question

What's the derivative of `f(x)=g(x)^(h(x))`?

Answer 1

`f'(x)=g(x)^(h(x)-1)(h'(x)g(x)ln(g(x))+h(x)g'(x))`

Explanation:

Using logarithmic differentiation:

`ln(f(x))=ln(g(x)^(h(x)))=h(x)ln(g(x))`

Differentiating both sides (chain, product rules):

`1/f(x)f'(x)=h'(x)ln(g(x))+h(x)1/g(x)g'(x)`

`1/f(x)f'(x)=(h'(x)g(x)ln(g(x))+h(x)g'(x))/g(x)`

Multiply both sides by `f(x)=g(x)^(h(x))`:

`f'(x)=(g(x)^(h(x))(h'(x)g(x)ln(g(x))+h(x)g'(x)))/g(x)`

`f'(x)=g(x)^(h(x)-1)(h'(x)g(x)ln(g(x))+h(x)g'(x))`