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I want to show what Δz ≈ (∂z/∂x)Δx + (∂z/∂y)Δy

Most places I've seen call this a definition, but it's not really that obvious. I know that it makes perfect sense geometrically, but I want a little more.

One way I thought of approaching it is to put a tangent plane at the point x0 y0 and show that going along x then along y is like cutting diagonally across to x,y.

Basically I need to show that f(x+Δx ,y+Δy) = f(x+Δx, y) +f(x, y+Δy) - f(x,y).

Unfortunately I'm not good at math, not good at proofs, tired, and a bit busy/lazy :), so I'm calling in the troops. Thanks!