Answer
$\displaystyle p' = \frac{-sinq(q^{2}+1)+ qcosq(q^{2}-1)}{(q^{2}-1)^{2}}$
Work Step by Step
$\displaystyle p = \frac{q*sinq}{q^{2}-1}$
$\displaystyle p' = \frac{(q^{2}-1)(sinq + q*cosq) - 2q^{2}sinq}{(q^{2}-1)^{2}}$
$\displaystyle p' = \frac{q^{2}*sinq+q^{3}*cosq -sinq -qcosq - 2q^{2}sinq}{(q^{2}-1)^{2}}$
$\displaystyle p' = \frac{-q^{2}sinq+q^{3}cosq -sinq -q*cosq}{(q^{2}-1)^{2}}$
$\displaystyle p' = \frac{-sinq(q^{2}+1)+ qcosq(q^{2}-1)}{(q^{2}-1)^{2}}$
