Question

Help me on Multivariable Calculus please
(I'll post the question under)

Answer 1

This is a Linear Approximation Formula.
\[\Delta f \approx f_x \Delta x + f_y \Delta y\]
I have no idea how to justify/prove this.

Answer 2

I meeeeaaan, I would just compare it to approximating a normal function:

Answer 3

You look at how the function is changing at a point, and you assume that that slope is constant. That gives you your approximation. How right or wrong your approximation is depends on how much that slope is actually changing.