How to find the slope of curves at origin if the derivative becomes indeterminate What's the general method to find the slope of a curve at the origin if the derivative at the origin becomes indeterminate. For Eg--

What is the slope of the curve $x^3 + y^3= 3axy$ at origin and how to find it because after following the process of implicit differentiation and plugging in $x=0$ and $y=0$ in the derivative we get $0/0$.

Actually this question has been asked by me before and a sort of satisfactory answer that I got was 

" For small $x$ and $y$, the values of $x^3$ and $y^3$ will be much smaller than $3axy$, so the zeroes of the function will be approximately where the zeroes of $0=3axy$ are -- that is, near the origin the curve will look like the solutions to that, which is just the two coordinate axes. So the curve will cross itself at the origin, passing through the origin once horizontally and once vertically.
(This is also why implicit differentiation can't work at the origin -- the solution set simply doesn't look like a straight line there under any magnification)." 

Edit

If I approximate the function by saying that at (0,0) , the behavior is dominated be 3axy term as x^3 and y^3 are very small and then 3axy=0 and then tangents are x=0 and y=0 . Is doing so (saying x=0 and y=0) linear Approximation only. Because I am approximating the curve with a straight line at origin . But linear Approximation is  1st derivative (1st term of Taylor series). This cannot be right because Taylor series can't be formed where derivative doesn't exist*.

And if this is right then the function is approximately given by 3axy=0 at (0,0). But how does this give the tangent at (0,0).How shall I go about ?

Edit:

Is the answer give  right because the solpe does exist.
It's as simple of that: No (simple) curve, no derivative, no slope.
Use l'Hopital's rule.$ $ This rule, the proof of which is very complex, states that if you get 0/0 or infinity/infinity for some limit function $h(x)=f(x)/g(x)$, you can take the derivative of the top and bottom functions and recalculate. Thus, you would do $f'(x)/g'(x)$. If this result is 0/0 or infinity over infinity, try the next derative, and the next, and so forth. In your particular case, it can be confusing because the implicit function contains both y and x, so which do you take the derivative with respect to? The correct way is to take the derivative with respect to x because you are ultimately solving for dy/dx, even if you use implicit. When differentiating with resepect to x, remember to treat y as a constant. In this case, you should get  6x/constant after one application of l'Hops, with x being 0. Thus, the slope at x=0 is indeed 0. ps. the derivative is "technically" undefined even though it can be determined, so this is just how to get the slope.