If $y=e^{\sin ^{-1}(t^{2}-1)}$ and $x=e^{\sec ^{-1}(\frac{1}{t^{2}-1})}$,then $\frac{dy}{dx}$ is equal to

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

If $x = \sin t$ and $y = \sin pt$,then the value of $(1 - x^2) \frac{d^2 y}{d x^2} - x \frac{d y}{d x} + p^2 y =$

If $x = 2\cos t - \cos 2t$ and $y = 2\sin t - \sin 2t$,then at $t = \frac{\pi}{4}$,find $\frac{dy}{dx}$.

The $2^{nd}$ derivative of $a \sin^3 t$ with respect to $a \cos^3 t$ at $t = \frac{\pi}{4}$ is

Let $t \in (0, 1)$ and $\alpha \in (0, \frac{\pi}{4})$. If $x = \text{cosec}^{-1} \left( \frac{1 + t^2}{2t} \right)$, $y = \cot^{-1} \left( \frac{\sqrt{1 - t^2}}{t} \right)$ and $\frac{dy}{dx} = f(t)$, then the value of $f(\tan \alpha)$ is