Orthogonal Trajectory Problem

1. Jul 16, 2006

mbaron

I am working on this problem, and have a simple question.

Determine the orthogonal trajectory of
$$x^p + Cy^p = 1$$
where p = constant.

I start out by taking the derivative with respect to x. My question is this. does
$$Cy^p$$ become $$Cpy^{p-1}$$ or $$C_1y^{p-1}$$ ?

Thanks,
Morgan

Last edited: Jul 16, 2006
2. Jul 16, 2006

HallsofIvy

C1?? There isn't any "C1" in your original formula!

The derivative of yp with respect to y is pyp-1. The derivative of Cyp with respect to y is Cpyp-1. By the chail law, the derivative of Cyp is $Cpy^{p-1}\frac{dy}{dx}$. Solve the resulting equation for $\frac{dy}{dx}$ to find the slope of the tangent line to the original trajectory at each point.

3. Jul 16, 2006

mbaron

If p is a constant and C is a constant isn't
$$C_1$$
just another constant? Isn't
$$C_1y^{p-1}\frac{dy} {dx}$$
the same as what you have?

Thanks for pointing out the chain rule, I missed that.