What's the derivative of $f (x) = {g (x)}^{h (x)}$ $f(x)=g(x)^(h(x))$ ?

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Answer 1 · 2016-09-20T21:09:09

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

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 $f(x)=g(x)^(h(x))$ :

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

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

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