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

1 Answer
Sep 20, 2016

#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))#