Induced gravitation action

Considering the action for induced gravity:

$$S=int d^4xsqrt{-g}left(epsilonphi^2 R-frac{1}{2}(partialphi)^2+V(phi)right).$$

I was trying to get the metric field equations by doing the usual procedure and for the $delta(phi^2 epsilon R)$ term, using the Leibniz rule twice, I got this:

$$ delta(phi^2epsilon R) = [epsilon phi^2 R_{munu} +g_{munu}square(epsilonphi^2) – nabla_{mu}nabla_{nu}(epsilonphi^2)] delta g^{munu} $$

Is this correct? Or do I have to do some type of chain rule for $phi^2$ in the Leibniz rules?