Partial Derivatives of Multi-Variable Functions

Question

Let $f(x)$ be a differential function and $F(x,y)=f(x-y)$. Show that $$ \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} = 0 $$

I have no idea where to even begin with this problem. A small hint to start would be a great help!

Answer 1

If $T(x, y) = x - y,$ then $T$ is linear and $T'(x,y) = T.$ By the chain rule, $F'(x, y) = f'(x - y) \circ T'(x, y) = f'(x - y) T$ (the latter being a scalar multiple of a linear function). Identifying $T$ with its canonical matrix $[1, -1],$ we reach $\partial_x F + \partial_y F = f'(x - y)(1 - 1) = f'(x - y) (0) = 0.$ Q.E.D.