calculating partial derivatives at the origin when they are not defined.

Question

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?

Answer 1

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.