I would like to solve this
$$\frac{\partial\int_{S(x)>S(\theta)}(S(x)-S(\theta))dF(x)}{\partial\theta}$$
where $S$ is a single-valued, differentiable, and strictly increasing function and $F$ is a distribution function.
This is more advanced than the Leibniz rule explained in wikipedia or any page that comes up easily. In those pages, range of integration is defined easily as interval.
More generally, I want to solve when there are more than 1 variable, but differentiation is taken by one variable:
$$\frac{\partial\int_{S(x_{1},x_{2})>S(\theta_{1},\theta_{2})}(S(x_{1},x_{2})-S(\theta_{1},\theta_{2}))dF(x_{1},x_{2})}{\partial\theta_{1}}$$
Does it need to be zero? At least that should be what I will be getting to be consistent with the paper I am reading. I tried to apply the concept in the Leibniz rule as it is defined online but it doesn't seem to be straightforward.