I want to know how I can make the derivative of this piecewise function respect to the X variable.
I know that in the point (0,0) you have to use the definition but I need the general derivative of all the function respect to X.
Thank you so much for the answers.
$
f(x,y)=
\begin{cases}
\frac {xy(x^2-y^2)}{x^2+y^2}, & \text{if (x,y) is not equal to (0,0)} \\
0, & \text{if (x,y) is equal to (0,0)}
\end{cases}
$
Best Answer
So you know how to find $f'_x(0,0)$, which is what most people usually have trouble with.
Finding $f'_x(x,y)$ for $(x,y) \neq (0,0)$ is the easy part: the values of $f$ near such a point are all given by the formula $xy(x^2-y^2)/(x^2+y^2)$ (and are not affected by the special value $f(0,0)=0$), so just take $\partial/\partial x$ of that expression.