1. ## differentiable

Hi!

I have this funtion: $\displaystyle {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}}$ and if $\displaystyle (x,y)=(0,0) \rightarrow f(x,y)=0$

so if I want to demonstrate it is differentiable in $\displaystyle (x,y)=(0,0)$, here is what I do:

1) I check if $\displaystyle f(x,y)$ is continuous in $\displaystyle (x,y)=(0,0)$.
2) I find partial derivatives.
3) I check if partial derivatives are continuous in (0,0).

In case steps 1) and 3) are affirmatives, the function is differentiable in $\displaystyle (x,y)=(0,0)$.

Too wrong?

So, in this case:

1)It is continuous.

2) I calculate partial derivate respect x: $\displaystyle \dfrac{\partial f}{\partial x}=\displaystyle\frac{x^{4}+3x^{4}y^{2}}{{(x^{2}+y ^{2})}^{2}}$

When I calculate the limit of partial derivate respect $\displaystyle x$ in (0,0) and it doesn't exit, so partial derivates are not continuos in $\displaystyle (0,0)$ so $\displaystyle f(x,y)$ is not differentiable,
Right?

Thank you very much! Great forum!

2. Why is continuous at (0,0) ? what is f(0,0) ?

3. I forgot to define it in (0,0), thank you!

4. I have corrected the solution, can you have a look?

Thank you!