Derivative of max function of a function

Question

I am trying to find a partial derivative of a max function that has another function inside. More specifically... \begin{equation} \frac{\partial}{\partial x} \max(0,a-\frac{x}{x+y}) \end{equation} where $a$ is a constant.

I guess I need to break up into intervals where $\frac{x}{x+y} \leq a$ and $\frac{x}{x+y} > a$ and get partial derivatives. However I am having troubles going forward.

Answer 1

It is $0$ when $\frac x {x+y} >a$ and $-\frac y {(x+y)^{2}}$ when $\frac x {x+y} <a$. When $\frac x {x+y} =a$ the derivative does not exist.