Answer
(a) $\frac{dy}{dx} = \frac{dsin~\theta}{r-d~cos~\theta}$
(b) The trochoid does not have a vertical tangent.
Work Step by Step
(a) We can find $\frac{dy}{dx}$:
$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{dsin~\theta}{r-d~cos~\theta}$
(b) Note that $\frac{r}{d} \gt 1$ when $r \gt d$
We can try to find values of $\theta$ such that the denominator of $\frac{dy}{dx}$ is $0$
$r-d~cos~\theta = 0$
$d~cos~\theta = r$
$cos~\theta = \frac{r}{d} \gt 1$
There are no values of $\theta$ such that $cos~\theta \gt 1$
Therefore, the trochoid does not have a vertical tangent.
