Orthogonal Trajectory Problem

mbaron

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

Determine the orthogonal trajectory of
[tex] x^p + Cy^p = 1 [/tex]
where p = constant.

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

Thanks,
Morgan

HallsofIvy

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

The derivative of y^p with respect to y is py^p-1. The derivative of Cy^p with respect to y is Cpy^p-1. By the chail law, the derivative of Cy^p is [itex]Cpy^{p-1}\frac{dy}{dx}[/itex]. Solve the resulting equation for [itex]\frac{dy}{dx}[/itex] to find the slope of the tangent line to the original trajectory at each point.

mbaron

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

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