The chain rule and logarithms practice question 29

Both the base $f(x^{2})$ and the exponent $x$ change with $x$. Which method handles a variable base raised to a variable power?

Take logarithms first: $\log g(x) = x\log f(x^{2})$. Differentiate the right side with the product rule, and $\log f(x^{2})$ with the chain rule, which here is used twice.

$\dfrac{d}{dx}\log f(x^{2}) = \dfrac{f'(x^{2})\cdot 2x}{f(x^{2})}$. Evaluate everything at $x = 1$, then multiply $g'(1)/g(1)$ by $g(1)$.