The chain rule and logarithms practice question 22

Both the base and the exponent of $f$ depend on $x$. Does the power rule or the rule for $a^{x}$ still apply, and what does a logarithm do to an exponent?

Neither rule applies when both the base and the exponent vary. Take $\log$ of both sides first: $\log f = x\log(x^{2} + 1)$.

Differentiate implicitly: $\dfrac{f'}{f} = \log(x^{2} + 1) + x\cdot\dfrac{2x}{x^{2} + 1}$. Then multiply back by $f(x)$.