Partial Derivatives extra credit question

[thearchress]

Hi, my question is about an extra credit question that my professor assigned us. The problem is:

I've gotten through and found that both partial derivatives are the same:

\(\displaystyle (x^6 + 9y^2x^4 - 9x^2y^4 - y^6)/((x^2+y^2)^3)\)

I'm not sure what I'm doing wrong, or if I'm missing some clue, because I don't see how they could not be equal when clearly the equations are the same.

Thanks!

 

[galactus]

Here's a hint.

Prove that \(\displaystyle f_{x}(0,0) \;\ \text{and} \;\ f_{y}(0,0)\) exist, but f is not continuous at (0,0). Show that \(\displaystyle f_{x}(0,0)=0 \;\ \text{and} \;\ f_{y}(0,0)=0\) by expressing these derivatives as limits.

To prove that f is not continuous at (0,0), show that \(\displaystyle \L\\\lim_{(x,y)\to\(0,0)}f(x,y)\) does not exist by letting \(\displaystyle (x,y)\rightarrow(0,0)\) along y=0 and y=x.

 

[thearchress]

Thanks! I had forgotten about taking the limit of the second derivative.