1. Differentiability for Functions of Two (or more!) Variables Local Linearity

2. Recall that when we zoom in on a “sufficiently nice” function of two variables, we see a plane.

3. Differentiability: A precise definition A function f(x,y) is said to be differentiable at the point (a,b) provided that there exist real numbers m and n and a function E(x,y) such that for all x and y and

4. What is meant by “sufficiently nice”? Suppose we zoom in on the function z=f(x,y) centering our zoom on the point (a,b) and we see a plane. What can we say about the plane? The partial derivatives for the plane at the point must be the same as the partial derivatives for the function--- The partial derivatives for the plane at the point must be the same as the partial derivatives for the function--- In particular, the partial derivatives must all exist! The equation for the tangent plane is The equation for the tangent plane is

5. Partial Derivatives Exist Suppose we have a function Notice several things: The function is not continuous at x=0. The function is not locally planar at x=0. Both partial derivatives exist at x=0.

6. Directional Derivatives? It’s not even good enough for all of the directional derivatives to exist! Just take a function that is a bunch of straight lines through the origin with random slopes. (One for each direction in the plane.)

7. Directional Derivatives? If you don’t believe this is a function, just look at it from “above.”

8. Directional Derivatives? If you don’t believe this is a function, just look at it from “above”. There’s one output (z value) for each input (point (x,y)).

9. Differentiability The function z = f(x,y) is differentiable (locally planar) at the point (a,b) if and only if the partial derivatives of f exist and are continuous in a small disk centered at (a,b).