Proof of a chain rule for partial derivatives

Simple proof of a basic chain rule for partial derivatives. The notion of differentiability is incorporated into the proof. Such ideas have lots of applications in university mathematics.

[nicholascousar]

Would you do a proof of the chain rule involving two intermediary variables each depending on two independent variables? For instance, let w be a differentiable function of x and y, and let x and y be differentiable functions of r and s. The only explanations I can find involve tree diagrams, and that seems to hand-wavey.

[花無缺-pf]

How to differentiate w with respect to x? x=g(r, s)

[Leo-iobq]

You didn't define the domain and co-domain. Could you please do that