Suppose I have a maximisation problem $$\underset{0\le R(y)\le y} \max E\left\{u_B\left[ y-R( y)\right]\right\}$$ subject to$$E\left\{u_L\left[R( y)\right]\right\}\ge \bar U_L$$ $B$ and $L$ are simply two people, with respective utility function $\bar U_L $ is just a constant.
$y$ is the possible return on a project, which should have a probability density function $f(y)$, while $R(y)$ is the amount of the return that goes to agent $L$.
To solve the question, I construct the Lagrangian equation $$\mathcal{L}=E\left\{u_B\left[ y-R( y)\right]\right\}+\lambda\left[E\left\{u_L\left[R( y)\right]\right\}-\bar U_L\right]$$ I, however, don't know how to calculate $$\partial \mathcal{L}\over \partial R(y)$$ I am not familiar with differentiating the expected value of a continuous function.