I am trying to find the derivative of the function $h(x)=f(x)^{g(x)}$. I just wanted to be sure my derivation was correct:
We proceed by using logarithmic differentiation.
$h(x)=f(x)^{g(x)}$
$\log (h(x))=g(x) \log (f(x))$
$\frac{h'(x)}{h(x)}=g'(x) \log (f(x))+\frac{g(x)f'(x)}{f(x)}$
Thus, $h'(x)=h(x)\left(g'(x) \log (f(x))+\frac{g(x)f'(x)}{f(x)}\right)$
Does this look correct?
$\endgroup$ 41 Answer
$\begingroup$In your next-to-last equation you have the prime mark on $f(x)$ in the wrong place. You corrected that in the last equation.
Your last equation is not expressed explicitly in terms of $f$ and $g$. You should replace the $h(x)$ on the right and simplify.
Your analysis assumes that $f(x)>0$ in the interval you are investigating. That is not stated in the beginning, so you should also have an analysis for the possibility $f(x)=0$.
Except for my first comment, those are weaknesses, not actual errors, so correct that next-to-last line and your work may be acceptable. That depends on your context.
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