Let $y = \sqrt[p]{x^3 y}$. If $\dfrac{dy}{dx} = \dfrac{3}{2}$ when $y = 1$, then the value of $p$ is equal to $\ldots$

If $f'(x) = \sin^2 x$ and $y = f\left(\dfrac{2x-1}{x^2+1}\right)$, then $\dfrac{dy}{dx}$ at $x = 1$ is

If $f : R \to R$ is an even function then

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