The log of a product is a sum of two logs - not a product of 2 logs.

So you don't want the product rule, though you do want to differentiate f and g. Your f is fine, but g dodgy.

Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!