Graduate Variation of the kinetic term in scalar field theory

[Baela]

Varying $\partial_\lambda\phi\,\partial^\lambda\phi$ wrt the metric tensor $g_{\mu\nu}$ in two different ways gives me different results. Obviously I'm doing something wrong. Where am I going wrong?

Method 1: \begin{equation}
(\delta g_{\mu\nu})\,\partial^\mu\phi\,\partial^\nu\phi
\end{equation}

Method 2: \begin{align}&\quad\,\, (\delta g^{\mu\nu})\,\partial_\mu\phi\,\partial_\nu\phi \nonumber \\
&=(-g^{\mu\rho}g^{\nu\sigma}\delta g_{\rho\sigma})\,\partial_\mu\phi\,\partial_\nu\phi \quad (\because \delta g^{\mu\nu}=-g^{\mu\rho}g^{\nu\sigma}\delta g_{\rho\sigma} \,\,\text{as can be checked by varying the identity}\,\, g^{\mu\lambda}g_{\lambda\nu}=\delta^\mu_\nu) \nonumber\\
&=-(\delta g_{\rho\sigma})\,\partial^\rho\phi\,\partial^\sigma\phi
\end{align}
The second result differs from the first one by a minus sign. What's going wrong?

 

[Orodruin]

In Method 1 you are missing the variations of the metric inside the definitions $\partial^\mu \phi = g^{\mu\nu}\partial_\nu \phi$.