Tangent at self-intersection point

[TPAINE]

Homework Statement

Find the slopes of the two tangent lines of x^3-y^2+x^2=0 at 0,0.

Homework Equations

Differentiating implicitly we get (dy(x))/(dx) = (x (2+3 x))/(2 y).

The Attempt at a Solution

I'm not sure how to deal with the derivative being undefined at 0,0 when there are clearly two tangent lines.

 

[LCKurtz]

Try solving the equation for y and differentiate each branch explicitly.

 

[TPAINE]

I'm aware it can be done that way. However, is there a way to do it without separating the curve into branches, for equations where that is not so easy (or impossible)?

 

[LCKurtz]

Good question. In the situation where you can't write an explicit formula for the function I don't think there is any guarantee you can do it. I could be wrong, but I don't think so.