An identity relating partial derivatives to a function

Question

I saw somewhere that given a function f(x,y) you can approximate with the following

Δf≈ ∂f/∂x * Δx + ∂f/∂y * Δy

Can someone explain why this works and where this came from? Thanks

Answer 1

Using Taylor on the first variable,

$$g(x,y):=f(x+\Delta x, y)\approx f(x, y)+\dfrac{\partial f(x,y)}{\partial x}\Delta x.$$

Then using it on the second variable, $$f(x+\Delta x,y+\Delta y)=g(x,y+\Delta y)\approx g(x,y)+\dfrac{\partial g(x,y)}{\partial y}\Delta y \\\approx f(x, y)+\dfrac{\partial f(x,y)}{\partial x}\Delta x+\dfrac{\partial f(x,y)}{\partial y}\Delta y+\dfrac{\partial^2f(x,y)}{\partial x\,\partial y}\Delta x\Delta y.$$

The last term is negligible.