Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate

$\frac{\partial}{\partial \alpha} \mathbb{E} [f(x)]$

Ultimately, this is for the following purpose: Suppose $\alpha^*(y)$ is defined by the equation

$\mathbb{E} [f(x) \;|\; \alpha] \equiv \alpha + y$

I am interested in studying the implicit function $\alpha^*(y)$ so defined. Any tips to characterize it would be greatly appreciated.

Suppose $x$ has a binomial distribution with chance $\alpha$ drawn $k$ times, and let $f(x)$ be a positive convex real valued function. I would like to evaluate

$$\frac{\partial}{\partial \alpha} \mathbb{E} [f(x)]$$

Ultimately, this is for the following purpose: Suppose $\alpha^*(y)$ is defined by the equation

$$\mathbb{E} [f(x) \mid \alpha] \equiv \alpha + y$$

I am interested in studying the implicit function $\alpha^*(y)$ so defined. Any tips to characterize it would be greatly appreciated.