Field equations involving a field potential term

We are given an action of the form:

$$S=int d^4xsqrt{-g}left(-frac14F_{munu}F^{munu}+V(B_{sigma}B^{sigma}) +Rlambda B_{mu}B^{mu}right)$$

where $R$ is the curvature scalar, $lambda$ is a constant, $F^{munu}=partial_nu B_mu -partial_mu B_nu$ is the field strength tensor and $V(B_{sigma}B^{sigma})$ is the field potential. Writing:

$F_{alphabeta}F^{alphabeta}=g^{alphalambda}g^{betarho}F_{alphabeta}F_{lambdarho}spacespacespacespacespacespace$ and $spacespacespace spacespacespacespacespacespace B_{mu}B^{mu}=B_{mu}B_{nu}g^{munu}$

we vary all the terms that have the metric in them in order to obtain the metric field equations, but how do I extract $delta g^{munu}$ from the $V(B_{sigma}B^{sigma})$ term? On the same token, if I want to find the field equations for the vector field $B_{mu}$ how can I extract a $delta B_{mu}$ out of the $V(B_{sigma}B^{sigma})$ term?

EDIT:

The equations I obtained were these:

$$ -frac{1}{2} g_{munu} mathfrak{L} + lambda[B_mu B_nu R+B_sigma B^sigma R_{munu}+g_{munu} square(B_sigma B^sigma) -nabla_mu nabla_nu(B_sigma B^sigma)]-frac{1}{2}F_{mualpha}F^{alpha}_{nu} +frac{partial V(B^2)}{partial B^2} B_mu B_nu =0 spacespacespacespacespacespacespace(1)$$

where:

$$ mathfrak{L} =-frac14F_{munu}F^{munu}+V(B_{sigma}B^{sigma}) +Rlambda B_{mu}B^{mu} $$

and:

$$ frac{delta V(B^2)}{delta(B^2)} frac{delta (B^2)}{delta g^{munu}} = frac{delta V(B^2)}{delta(B^2)} {B_nu B_mu} delta g^{munu}$$

This term from the integral gives us the final term in equation (1) which I converted the variational derivative into a partial derivative. Not sure if this is the right approach