If $x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$,then $\left(\frac{d^2 y}{dx^2}\right)_{\theta=\pi / 2}=$

If $x=a(\cos t+t \sin t)$ and $y=a(\sin t-t \cos t)$,then $\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}=$

If $x = at^2$ and $y = 2at$,then find $y_2$ (where $t \neq 0$).

If $x = 2 \sqrt{2} \sqrt{\cos 2 \theta}$ and $y = 2 \sqrt{2} \sqrt{\sin 2 \theta}$,$0 < \theta < \frac{\pi}{4}$,then the value of $\frac{dy}{dx}$ at $\theta = 22 \frac{1}{2}^{\circ}$ is

If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{dy}{dx}$ for $x=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t$.

If $x=f(t)$ and $y=g(t)$,then the value of $\frac{d^{2} y}{d x^{2}}$ is