Polchinski uses the graviton-dilaton action (8.1.9) in his String Theory book $$S_1= \frac{1}{2\kappa_0^2}\int d^D x\, \sqrt{-G} e^{-2\Phi} \left[ {R} + 4 \nabla_\mu\Phi \nabla^\mu \Phi \right] \tag{8.1.9} $$ for the Kaluza-Klein theory. He then rewrites this in terms of the Kaluza-Klein fields as $$S_1= \frac{\pi R }{\kappa_0^2}\int d^d x\, \sqrt{-G_d} e^{-2\Phi+\sigma } \left[ {R_d} -4\partial_\mu\Phi \partial\sigma + 4 \partial_\mu\Phi \partial^\mu \Phi-\frac{1}{4} e^{2\sigma}F_{\mu\nu}F^{\mu\nu} \right] $$ I understand where most of this comes. The integration over $x^d$ gives $2\pi R$. We have $ \sqrt{-G}= e^\sigma \sqrt{-G_d}$. We use (8.1.8), i.e. $$ R = R_d -2e^{-\sigma} \nabla^2 e^\sigma-\frac{1}{4} e^{2\sigma} F_{\mu\nu}F^{\mu\nu} $$ and we replace $\nabla_\mu \Phi$ by $\partial_\mu \Phi$ as $\Phi$ is a space-time scalar (I assume this is correct?).
But how does the $-2e^{-\sigma} \nabla^2 e^\sigma$ change into $-4\partial_\mu\Phi \partial\sigma $? What am I missing?
