(x^2) - xy + (y^3) = 10 has a vertical tangent when x = ?




I believe I have to take the derivative, by implicit differentiation, but I don't know what to do next. Or I could be wrong..

You're right. Implicit differentiation.

x^{2}-xy+y^{3}=10

2x-(x\cdot\frac{dy}{dx}+y)+3y^{2}\cdot\frac{dy}{dx}=0

Solve for dy/dx


There's a little trick you can use with implicit diff to make it a little easier.

Find the derivative wrt x. You get 2x-y... [1]

The derivative wrt y. You get 3y^{2}-x... [2]

Now, just use \frac{-[1]}{[2]}. You'll get the same thing.