If $x=a(1-\cos \theta)$ and $y=a(\theta+\sin \theta)$,then $\frac{dy}{dx} =$ . . . . . .

If $x = 2 \sin \theta - \sin 2 \theta$ and $y = 2 \cos \theta - \cos 2 \theta$ where $\theta \in [0, 2 \pi]$,then find the value of $\frac{d^{2} y}{dx^{2}}$ at $\theta = \pi$.

For the curve $y = 3 \sin \theta \cos \theta$,$x = e^{\theta} \sin \theta$,$0 \leq \theta \leq \pi$,the tangent is parallel to the $x-$axis when $\theta$ is

The slope of the tangent to the curve $x = 3t^2 + 1, y = t^3 - 1$ at $x = 1$ is

Consider the curve represented parametrically by the equations $x = t^3 - 4t^2 - 3t$ and $y = 2t^2 + 3t - 5$,where $t \in \mathbb{R}$. If $H$ denotes the number of points on the curve where the tangent is horizontal and $V$ denotes the number of points where the tangent is vertical,then:

If $x = \sin 2\theta \cos 3\theta$ and $y = \sin 3\theta \cos 2\theta$,then find $\frac{dy}{dx}$.