I couldn't follow your equation's progression because you didn't explain anything except your variable declarations and your final derivatives.
This seems like interestin stuff for someone into physics.
I was just mostly confused because I havent taken physics in years and seeing equations just appear upsets me as a rule.
for example I do not know what any of the original formulas were which you derived. That would be helpful to someone reading this so that they could see where you started and how you progressed. There is no shame in showing too much work.