I am researching a supply chain coordination method through contract design, and I got stuck in a point where I have to make a partial derivative of a definite integral. However, the variable being derivated is only present on the limits of the integral.
I need to derivate the following equation in terms of the price $p$
$$q\int_{q/S_0}^{q/d}G(\xi)d\xi$$
and
$$d = a - bp + ke$$
$G(\xi)$ is a cumulative distribution function of a random variable.
Since the price is only present inside $d$ in the upper limit of the integral I am not sure how it can be derivated in terms of the price $p$.
The derivative is $qG(\frac q d)\frac d {dp} (\frac q d)$ which is $G(\frac q d) (-\frac q {d^{2}}) (-b)$ by Chain Rule.