Answer
The Derivative is:
$\frac{ds}{dt}=\frac{-1}{1-\cos t}$
Work Step by Step
$s=\frac{\sin t}{1-\cos t}$
Applying Derivative rules:
$\frac{ds}{dt}=\frac{(1-\cos t)\cdot\frac{d}{dt}(\sin t)-(\sin t)\cdot\frac{d}{dt}(1-\cos t)}{(1-\cos t)^2}$
$\frac{ds}{dt}=\frac{(1-\cos t)(\cos t)-(\sin t)(0+\sin t)}{(1-\cos t)^2}$
$\frac{ds}{dt}=\frac{\cos t-\cos^2t-\sin^2 t}{(1-\cos t)^2}$
$\frac{ds}{dt}=\frac{\cos t-\cos^2t-\sin^2 t}{(1-\cos t)^2}$
$\frac{ds}{dt}=\frac{(-1)(-\cos t+\cos^2t+\sin^2 t)}{(1-\cos t)^2}$
Applying trigonometric identities: $\cos^2 t+\sin^2 t=1$
$\frac{ds}{dt}=\frac{(-1)(1-\cos t)}{(1-\cos t)(1-\cos t)}$
$\frac{ds}{dt}=\frac{-1}{1-\cos t}$
