Differentiation Problem

This is a differentiation problem that I stumbled across while trying to calculate log-likelihood estimate, so it is related to statistics.

I'm having problems with the differentiation below

I can see how the
is obtained using the product rule, but I don't see how the term after that,
, was obtained. Shouldn't that term be
instead? See how I got this term. What have I done wrong?

You are differentiating with respect to sigma^2, not sigma. It would be easier if you let t=sigma^2 in the logL, and get the partial derivative wrt t. :)