Question

Logarithmic differentiation (see inside)

Answer 1

\[
\Large f(x)=(\sin x)^{\cos x}+(\cos x)^{\sin x}
\]
He wants me to use logarithmic differentiation, which is to take \((\log \circ f)'(x)\cdot f(x)\) to find \(f'(x)\). I feel like I'm missing something, because it seems like it would be pretty easy to just rewrite as \(\Large f(x)=e^{\cos x \log \sin x}+e^{\sin x \log \cos x}\) and derive regularly. Am I missing a trick here?

Answer 2

The other questions in this section have been things like \(f(x)=\dfrac{(3-x)^{1/3}x^2}{(1-x)(3+x)^{2/3}}\) where taking the log made manipulating it way easier.