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) * [tex] \frac{1}{sin^{2}(x)} [/tex] - sin(x) * [tex] \frac{cos(x)}{sin(x)} [/tex]
r' = [tex] \frac{-cos(x)}{sin^{2}(x)} [/tex] - cos(x)
There are other forms. Good luck!
