Answer
$\frac{dq}{dt}$ = $\frac{[sint - t(cost)]}{t^{2}}$ $csc^{2}(\frac{sint}{t})$
Work Step by Step
$q$ = $cot$($\frac{sint}{t}$)
differentiate with respect to x
$\frac{dq}{dt}$ = $\frac{d}{dt}$[$cot$($\frac{sint}{t}$)]
solve derivatives using the chain rule for
$\frac{dq}{dt}$ = -$csc^{2}(\frac{sint}{t})$ $\frac{d}{dt}$[$\frac{sint}{t}]$
$\frac{dq}{dt}$ = -$csc^{2}(\frac{sint}{t})$ $\frac{[(t)\frac{d}{dt}(sint) - (sint)\frac{d}{dt}(t)]}{t^{2}}$
$\frac{dq}{dt}$ = -$csc^{2}(\frac{sint}{t})$ $\frac{[(t)(cost) - (sint)(1)]}{t^{2}}$
$\frac{dq}{dt}$ = $\frac{[sint - t(cost)]}{t^{2}}$ $csc^{2}(\frac{sint}{t})$
simplify
$\frac{dq}{dt}$ = $\frac{[sint - t(cost)]}{t^{2}}$ $csc^{2}(\frac{sint}{t})$
