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

If $x = a \sec^{2} \theta$ and $y = a \tan^{2} \theta$,then find $\frac{d^{2} y}{d x^{2}}$.

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

If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{dy}{dx}$ for $x = \cos \theta - \cos 2\theta$ and $y = \sin \theta - \sin 2\theta$.

If $x = at^2$ and $y = 2at$,then find $\frac{d^2 x}{dy^2}$.

The function $y(x)$ represented by $x=\sin t$,$y=a e^{t \sqrt{2}}+b e^{-t \sqrt{2}}$,$t \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$ satisfies the equation $(1-x^2) y^{\prime \prime}-x y^{\prime}=k y$,then the value of $k$ is