The standard method is to take natural logs of both sides and then differentiate...

However, - Wolfram|Alpha does it with the chain rule for two inner functions...

... which doesn't mean we have to, of course, but I fancy this picture makes it palatable...

... where

... is the double version of...

... the ordinary chain rule. As with that, straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed lines similarly but with respect to the (corresponding) dashed balloon expression which is (one of) the inner function(s) of the composite expression.

Shading shows that an expression has been treated as a constant on the way down (during differentiation).

You do, admittedly, need logarithmic differentiation to see the logic of the double-dashed differentiation, i.e. the logic of...

_____________________________________

Don't integrate - balloontegrate!

Balloon Calculus: Standard Integrals, Derivatives and Methods

Balloon Calculus Drawing with LaTeX and Asymptote!