Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 27K / b / Solution

Symmetry of $Q$ gives ${\partial }_{y_i}{e}^{y^{T}Qy/2}=e^{y^{T}Qy/2}{\sum }_j{q}_{ij}{y}_j$. Also ${\partial }_{y_i}f(i,y^2/2)=y_i{\partial }_{{\ell }_i}f$. Substitute these identities into the left side and integrate the resulting total $y_i$ derivative by parts. Compact support removes the boundary term and leaves exactly the right side.