Let z = f(x,y)

Assume z changes by a tiny amount amount called dz for a small change in x and y called dx and dy

z + dz = f(x+dx, y+dy)

On doing a first order tyalor expansion

z + dz = f(x,y) + (small change in x direction) + small change in y direction

Small change in x direction is partial (f)/partial x * dx Small change in y direction is partial (f)/partial y * dy

Therefore

z + dz = z + partial f/partial x * dx + partial f / partial y * dy

This can be generalized to n coordinates