In chapter 12.1 from Peskin & Schroeder we see how we can integrate out the high momenta of our theory. Here we consider a $\phi^4$ theory for which we can seperate high and low momenta modes as follows $$ S = \int{d^dx\left( \frac{1}{2}(\partial_\mu\hat{\phi})^2+\frac{1}{2}m^2\hat{\phi}^2+\lambda\left(\frac{1}{6}\phi^3\hat{\phi}+\frac{1}{4}\phi^2\hat{\phi}^2+\frac{1}{6}\phi\hat{\phi}^3+\frac{1}{4!}\hat{\phi}^4\right)\right)} $$

where the above is only the part of the action which depends on the high momenta denoted by $\hat{\phi}$. My question is on equation (12.17) which states that when we take a Taylor expansion of the interaction terms(with coefficient $\lambda$) we will have a term which is of the form $$ -\frac{1}{4}\int{d^dx\eta\phi^2(\partial_\mu\phi)^2} $$ where $\eta$ is the factor we get when we integrate the high momenta $\hat{\phi}$ out. From the given action I don't understand how one can have $\partial_\mu\phi$ in their Feynman diagrams.

In chapter 12.1 from Peskin & Schroeder we see how we can integrate out the high momenta of our theory. Here we consider a $\phi^4$ theory for which we can seperate high and low momenta modes as follows $$ S = \int{d^dx\left( \frac{1}{2}(\partial_\mu\hat{\phi})^2+\frac{1}{2}m^2\hat{\phi}^2+\lambda\left(\frac{1}{6}\phi^3\hat{\phi}+\frac{1}{4}\phi^2\hat{\phi}^2+\frac{1}{6}\phi\hat{\phi}^3+\frac{1}{4!}\hat{\phi}^4\right)\right)} $$

where the above is only the part of the action which depends on the high momenta denoted by $\hat{\phi}$. My question is on equation (12.17) which states that when we take a Taylor expansion of the interaction terms  (with coefficient $\lambda$) we will have a term which is of the form $$ -\frac{1}{4}\int{d^dx\eta\phi^2(\partial_\mu\phi)^2} $$ where $\eta$ is the factor we get when we integrate the high momenta $\hat{\phi}$ out. From the given action I don't understand how one can have $\partial_\mu\phi$ in their Feynman diagrams.