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At what distance along the central axis of a ring of radius R and uniform charge is the magnitude of the electric field due to the ring's charge maximum?

Now, I know that the equation for this problem is E = [k|qz|] / [(z^2+R^2)^3/2], which is the electric field magnitude of a charged ring. And to get the maximum, you have to find where the first derivative of the equation where it equals zero.

But when I take the derivative of the equation, I get a complete mess not even close to the answer. Can anyone give me some hints or a push in the right direction? Is my thinking incorrect? Thanks. :)