What are the Partial Derivatives at the Origin (0,0)?

[raphile]

Homework Statement

Evaluate the partial derivatives ∂f/∂x and ∂f/∂y at the origin (0,0), where:

f(x,y) = ((xy)^3/2)/(x^2 + y^2) if (x,y) ≠ (0,0);
and f(0,0) = 0.

Homework Equations

∂f/∂x(x0,y0) = lim(h->0) [(f(x0+h, y0) - f(x0, y0)) / h]

∂f/∂y(x0,y0) = lim(h->0) [(f(x0, y0+h) - f(x0, y0)) / h]

The Attempt at a Solution

When I tried to use the definition for ∂f/∂x I got lim(h->0) (0/h^3), and the same result for ∂f/∂y. Is it correct to say therefore the partial derivatives at the origin are zero?

 

[HallsofIvy]

Yes, the partial derivative with respect to x at (0,0) is taken as x= h approaches 0 and y= 0. That means that the numerator is is 0 for all h so the "difference quotient" is 0 for all non-zero h. The limit, as h goes to 0, is 0 so the partial derivative there is 0.

The same is true for the partial derivative with respect to y at (0,0).