I'm having trouble understand Eq 3.51 Lemma 3.3 in https://arxiv.org/pdf/2001.06188.pdf
The basic premise is
$$\begin{align} &\eta _{j}(t)=t^{1/2}\eta _{j}+(1-t)^{1/2}q_{n}^{1/2}\gamma _{j}, \;t \in [0,1], \notag \\&\eta _{j} =n^{-1/2}\sum_{\alpha =1}^{n}X_{j\alpha }x_{\alpha n}+b_{j}, \label{eetaj} \end{align}$$ where $X$ is $n\times n$ matrix with iid standard gaussian entries, $b$ is $n\times1$ with iid N$(0,\sigma^2)$ entries and $\gamma$ is $n\times 1$ vector with iid N$(0,1)$ entries. Define $$K_{jj}(t)=(\varphi ^{\prime}(\eta _{j}(t)))^{2} $$ so that $K$ is a diagonal matrix. Further define $$F_{j}(z,t)=(XS\mathcal{G}(z,t)SX^{T}\dot{K}(t))_{jj}.$$ where $$\mathcal{G}(z,t)=(SX^TK(t)XS-z I)^{-1}$$ Now, the author says that using the formula $E[Z f(Z)] = E[f'(Z)]$ for std normal random variables, we get the following $$\mathbf{E}\{F_{j}(z,t)(t^{-1/2}\eta _{j}-(1-t)^{-1/2}q_n^{1/2}\gamma _{j})\}=(tn)^{-1/2}\sum_{\alpha =1}^{n}\mathbf{E}\left\{ \frac{\partial F_{j}% }{\partial X_{j\alpha }}\right\} x_{\alpha n}$$ where the partial derivative denotes the "explicit" derivative (not applicable to $X_{j\alpha }$ in the argument of $K$ and $\dot{K}$. I've verified the identity about std normal RVs but I couldn't prove this equality. One idea was to condition on $\eta_j(t)$, however, then the RVs inside the expectation won't be standard normal anymore. Any help is appreciated.
