Why does Partial Derivative "Fail" despite Limit Definition Many posts here have answered questions regarding the partial derivatives of multivariable functions that are continuous at some point. 
Consider the equation $(xy)^\frac{1}{3}$ which is continuous at $(0,0)$. When we take the partial derivatives, we see that they yield an undefined result, so we need to resort to the limit definition. I suppose this makes sense since the limit definition will yield the partial derivative without possibly being at the point (it does not even require continuity for that matter).
For the above example, the limit definitions yield $0$ for both.
$\dfrac {\partial f}{\partial x} = \frac{y^\frac13}{3x^\frac{2}{3}}$ and $\dfrac {\partial f}{\partial y} = \frac{x^\frac13}{3y^\frac{2}{3}}$
Even though the limit definition solves the problem, I still find it difficult to see that taking the partial derivative through differentiation and evaluating is not exactly equivalent to finding the partial derivative at a point by the limit definition. Surely the limit definition provides the framework for taking the partial derivatives the "normal" way. Why does this "discrepancy" occur?
 A: The problem in this particular example is that you have used a differentiation rule incorrectly.  To get $$\frac{\partial f}{\partial x}=\frac{y^{1/3}}{3x^{2/3}},$$ I'm guessing you used the following rule: if $g(x)=ch(x)$ for some constant $c$ and for all $x$, then $g'(x)=ch'(x)$.  (Here $y$ is fixed, $c=y^{1/3}$, $g(x)=f(x,y)$, and $h(x)=x^{1/3}$.)  However, this rule is not literally correct.  The correct rule is that if $h'(a)$ exists and $g(x)=ch(x)$ for all $x$ (in a neighborhood of $a$), then $g'(a)$ exists and $g'(a)=ch'(a)$.  For $a=0$ and $h(x)=x^{1/3}$, $h'(a)$ does not exist, so you cannot apply the rule and conclude that your formula for $\frac{\partial f}{\partial x}$ is correct at $x=0$.
(Note that if $c\neq 0$, then the naive rule $g'(x)=ch'(x)$ is valid in the sense that for each $x$ either both derivatives exist and the equation is true or neither derivative exists.  That is because if $g'(a)$ exists for any particular $a$, you can write $h(x)=c^{-1}g(x)$ and then use the rule with $c^{-1}$ in place of $c$ to conclude that $h'(a)$ exists and $h'(a)=c^{-1}g'(a)$.  But when $c=0$, it is possible that $h'(a)$ is not defined but $g'(a)$ is, and that is exactly what is happening in this example, since $c=y^{1/3}$ is $0$ when $y=0$.)
