I would like to have Mathematica find the first and second derivatives of six functions I define and simplify the output as much as possible. The functions are simply the conversions between Cartesian and radar coordinates. Radar coordinates are just spherical coordinates with the polar angle instead measured positively from the x-y plane towards the z axis. I have tried quite a number of methods with no success to simplify the expressions that can be clearly simplified. It would also be nice if I could replace all occurrences of x^2+y^2+z^2 with r.

I have attached a cleaned up version of my progress so far.

Please note: to actually use the radar derivatives (velocity and acceleration), they must be plugged into the equations obtained by taking the derivative of r-vector and v-vector (e.g. velocity is v-vector = {rDot, -r thetaDot, r phiDot Cos[theta]}). It would be pretty neat if you know how to make Mathematica obtain these as well; I did them by hand.

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