Partial Derivative Homework

1. Oct 15, 2008

klopez

1. The problem statement, all variables and given/known data

Show that the function is not differentiable at (0,0).

f(x,y) = [ (xy)/(x2 + y2)(1/2) if (x,y) =/ (0,0)

[ 0 if (x,y) = (0,0)

3. The attempt at a solution

I know that the partial derivatives at point (0,0) = 0, so I don't know why the function is not differentiable at (0,0). Is there a certain equation that will help me prove that?

Thanks
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Oct 16, 2008

HallsofIvy

Staff Emeritus
You titled this "partial differential homework" but it is important to understand that has very little to do with partial derivatives. "Differentiable" is NOT a matter of having partial derivatives.

I think it is really important that you look up the definition of "differentiable" for functions of two variables. In Calculus of one variable, we typically define the "derivative" as a limit and then say that a function is "differentiable" if and only if that limit exists. In Calculus of more than one variable, it is standard to define "differentiable" separately from just the partial derivatives.

I know several equivalent definitions of "differentiable" for two variables but I don't know which one your textbook is using: look it up please.

3. Oct 16, 2008

tiny-tim

Hi klopez!

It's because existence of partial derivatives ∂f/∂x and ∂f/∂y only prove differentiablity in the x and y directions.

Hint: try some other direction.