# Tangent Lines and Implicit Function

I am asked to find the equations of the tangent lines at three different points for the following function:

$$y^{5}-y-x^{2}=-1$$

I am provided with a graph/sketch of the function and asked to find the tangent lines to the three different points at $x=1$. I found the following with implicit differentiation:

$$\frac{dy}{dx}=\frac{2x}{5y^{4}-1}$$

My problem is finding the points where the curve intersects $x=1$. It is pretty clear from the graph (though not explicitly marked) that the points will be $y=-1$,$y=0$, and $y=1$. Wolfram Alpha confirms this http://www.wolframalpha.com/input/?i=y%5E5-y-x%5E2%3D-1+and+x%3D1. I just don't know if seeing them visually is enough or if I have to derive the intersection points mathematically.

Thank you very much.

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Well, have you tried solving? When $x=1$, you have $y^5 - y -(1)^2 = -1$, which means $y^5 = y$, hence either $y=0$ or $y^4 = 1$, which means $y=\pm 1$. Satisfied?
Just so you know that in general there are also the complex solutions $\pm i$ but I don't think they were appropriate in the context. – Patrick Da Silva Nov 3 '11 at 3:36