Logarithmic Differentiation

Sometimes it's just easier to take the derivative of $\ln(y)$ than of $y$. In those cases, we can get $y'$ indirectly by using the chain rule. In fact, $$\frac{d}{dx} \Bigl(\ln(y)\Bigr)= \frac{\frac{dy}{dx}}{\quad y\quad },$$ or, equivalently $$\frac{dy}{dx} = y \frac{d}{dx}\Bigl(\ln(y)\Bigr).$$

So, to compute $\displaystyle \frac{dy}{dx}$, first compute $\ln(y)$, then take its derivative, and then multiply the answer by $y$. This procedure isn't always helpful, and in some cases it can make computations much harder, but in others it can make them much easier.

When should you use logarithmic differentiation?

— Whenever $\ln(y)$ is easier to differentiate than $y$!

If $y$ involves a bunch of powers, products and quotients, then $\ln(y)$ is likely to be simpler, thanks to the three basic rules of logs: $$\ln(ab) = \ln(a)+\ln(b), \qquad \ln\left(\frac{a}{b}\right)=\ln(a)-\ln(b), \qquad \ln(a^r) = r \ln(a).$$ On the other hand, if $y$ is a sum of terms, then $\ln(y)$ is likely to be a mess, since there's no simple rule for $\ln(a+b)$. In other words, your mileage will vary.