You must use product rule:
(dr/dx) = cos(x) dx/dr(cot(x)) + (dx/dr)cos(x) cot(x)
r' = cos(x) -csc²(x) + -sin(x) cot(x)
If simplification is needed:
r' = -cos(x) \(\frac{1}{sin^{2}(x)}\) - sin(x) \(\frac{cos(x)}{sin(x)}\)
r' = \(\frac{-cos(x)}{sin^{2}(x)}\) - cos(x)
There are other forms. Good luck!
