I'm currently working my way through an economics paper which has a derivative that at the moment I'm failing to see. I was hoping that it would be a simple case of the implicit function theorem however, using that approach I don't find the correct answer. If someone could suggest what I'm doing wrong it would be greatly appreciated. I apologise if this seems trivial.
We are given the equation \begin{equation} c^*=\frac{b(e)y+a\int_{0}^{c^*} cdG(c)}{r+b(e)+aG(c^*)} \end{equation} where $G$ is a distribution function. The goal is to compute the derivative \begin{equation} \frac{dc^*}{de}=\frac{(y-c^*)b'(e)}{r+b+aG} \end{equation} To my mind I would use the implicit function theorem to compute \begin{equation} \frac{dc^*}{de}=-\frac{\partial H}{\partial e}\bigg/\frac{\partial H}{\partial c^*} \end{equation} where $H$ is given by \begin{equation} H=c^*-\frac{b(e)y+a\int_{0}^{c^*} cdG(c)}{r+b(e)+aG(c^*)}=0 \end{equation} Also using the Leibniz integral rule \begin{equation} \frac{\partial}{\partial c^*}\int_{0}^{c^*} cdG(c)=G(c^*) \end{equation} Does this look like the correct strategy?