Solving a Calculation Problem - Basic Math/Stress Energy Tensor

[kalish]

Hello, I have a basic calculation problem. I have to find a term in the stress energy tensor from a lagrangian, that has many terms. I found the correct equation at first but know I think I made a mistake in calculation.

here is the term that make me problem, I have to find $$\frac{\delta(\sqrt{-g}\partial_\alpha \phi \partial^\alpha \phi)}{\delta g^{\mu\nu}} = \sqrt{-g}( \partial_\mu \partial_\nu \phi -\frac{1}{2}g_{\mu\nu}\partial_\alpha \phi \partial^\alpha \phi)$$

indeed that's what I found. BUT I found it using $$\frac{ \delta g^{\alpha\beta}}{\delta g^{\mu\nu}} \sqrt{-g}\partial_\alpha \partial_\beta \phi = \delta^\alpha_\mu \delta^\beta_\nu \sqrt{-g}\partial_\alpha \partial_\beta \phi$$

but now I strongly believe that
$$\frac{ \delta g^{\alpha\beta}}{\delta g^{\mu\nu}} \sqrt{-g}\partial_\alpha \partial_\beta \phi = (\delta^\alpha_\mu \delta^\beta_\nu + \delta^\alpha_\nu \delta^\beta_\mu ) \sqrt{-g}\partial_\alpha \partial_\beta \phi$$

that means twice the first term, and that is a problem as my final result will look like

$$\sqrt{-g}( 2\partial_\mu \partial_\nu \phi -\frac{1}{2}g_{\mu\nu}\partial_\alpha \phi \partial^\alpha \phi)$$

So where is the problem please? I am a little ashamed, I thought about it before but I wasn't convinced as I found the good result.
This is not homework as I have to make a calculation that has already been made during my training, not quite a homework.

 

[kalish]

Hello, I found my mistake.