The curve with equation y2 = 5x4 − x2 is called a kampyle of eudoxus. find an equation of the tangent line to this curve at the point (−1, 2).

To find the tangent line we will need the slope of the tangent at x=-1 (the x-coordinate of the point given). We find the slope by using the derivative of the curve.

Th curve given is  which can be solved for y by taking the root of both sides. We obtain 

We find the derivative using the chain rule. Bring down the exponent, keep the expression in the parenthesis, raise it to 1/2 - 1 and then take the derivative of what is inside.



Next we evaluate this expression for x=-1 and obtain:


So we are looking for a line through (-1,2) with slope equal to -9/2. We use y=mx+b with m=-9/2, x=-1 and y=2 to find b.


2-(9/2)=b
b=-5/2

So the tangent line is given by y=(-9/2)x+(-5/2)

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