What I have to do is to get a value of partial derivative at a $(0 , 0)$ point in a following function $\\ f(x,y) = \sqrt[3]{xy} \\ \\$
What I struggle with is that I get different results while using different methods. When I use definition I get $\frac{\partial f}{\partial x}(0, 0) = \lim_{x \to 0} \frac{f(x, 0) – f(0, 0)}{x} = 0\\ \\$, however I can clearly see that $\frac{\partial f}{\partial x} = \frac{y^{\frac{1}{3}}}{3x^\frac{2}{3}}$ but then x can't be equal to $0$ so the derivative shouldn't have any value at a $(0, 0)$ point. Can someone point out my mistake? Which method is right?
Best Answer
The limit is the definition. It always gives the right answer, because it is the right answer.
The other approach gives the correct partial derivative for most of the plane, but the fact that the resulting expression is undefined does not let you conclude that the partial derivative doesn't exist.