How do you show that the partial derivative w.r.t. $x$ of $z(x,y)$ is equal to the inverse of the partial derivative of $x$ with respect to $z$?

Question

For a general function $z(x,y)$, writing the derivatives as ratios of finite-differences, how do you show that:

is true? Please push me in the right direction of what to do. I know how to take partial derivatives and what finite differences are, but I don't understand this question. How do you even start? What are the steps involved? Please help. Thank you very much.