For a variation of the metric $g^{\mu\nu}$ with respect to $g^{\alpha\beta}$ you might expect the result (at least I did):

\begin{equation}
\frac{\delta g^{\mu\nu}}{\delta g^{\alpha\beta}}= \delta^\mu\_\alpha\delta^\nu\_\beta.
\end{equation}

but then to preserve the fact that $g^{\mu\nu}$ is symmetric under interchange of $\mu$ and $\nu$ we should probably symmetrise the right hand side like this:

\begin{equation}\frac{\delta g^{\mu\nu}}{\delta g^{\alpha\beta}}= \delta^\mu\_\alpha\delta^\nu\_\beta + \delta^\mu\_\beta\delta^\nu\_\alpha.\end{equation}

Is this reasonable/correct? If not, why not?

It seems that I can derive some weird results if this is right (or maybe I'm just making other mistakes).