So I am asked to find partial derivatives of the function $f(x,y) = (xy)^{\frac{1}{3}}$ at $(0,0)$. This is a past paper exam question.
I obtain $$\frac{\partial f}{\partial x} = \frac{y}{3(xy)^{2/3}}$$. This is undefined at the origin?
Yet in the solutions he said it is zero?
The formula you obtained only shows that the derivative is likely to not be continuous at $(0,0)$, but you can't evaluate it at $(0,0)$.
You have $$ \frac{\partial f}{\partial x}(0,0)=\lim_{h\to0}\frac{f(h,0)-f(0,0)}h=\lim_{h\to0}\frac{0-0}h=0. $$ Similar for the other partial derivative.
