How to apply the chain rule inside the expectation operator?

Question

I'm working with a function that involves a term $$ E_t\left[ V(w_{t+1})^{1-\gamma}\right]^{\frac{1-\eta}{1-\gamma}} $$ I need to differentiate this with respect to $w_{t+1}$. How should I apply the chain rule inside the expectation operator? I'm not sure if it should be $$ (1-\eta)E_t\left[ V(w_{t+1})^{1-\gamma}\right]^{\frac{\gamma-\eta}{1-\gamma}}E_t\left[V(w_{t+1})^{-\gamma}V'(w_{t+1})\right] $$ or $$ (1-\eta)E_t\left[ V(w_{t+1})^{1-\gamma}\right]^{\frac{\gamma-\eta}{1-\gamma}}E_t\left[V(w_{t+1})^{-\gamma}\right]E_t\left[V'(w_{t+1})\right] $$ or possibly something else?