- #1

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Determine the orthogonal trajectories of the given family of curves.

[tex] y = \sqrt{2\ln{|x|}+C} [/tex]

This is what I've done so far:

[tex] y = (2\ln{|x|}+C)^\frac{-1}{2} [/tex]

[tex] y' = -1/2(2\ln{|x|+C)(2/x) [/tex]

Now I understand to find the orthogonal lines I need to divide -1 by whatever I get, the problem is, I can't simplify this derivative.

I've messed around with it a bit, and I have this:

[tex] -(2\ln{|x|}+C)/x [/tex]

How else can I simplify this?

Thanks.