If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{dy}{dx}$ for $x=4t$ and $y=\frac{4}{t}$.

If $f^{\prime}(x)=\sqrt{2 x^2-1}$ and $y=f(x^3)$,then find the value of $\frac{dy}{dx}$ at $x=1$.

Find $\frac{dy}{dx}$,if $x = at^2$ and $y = 2at$.

The angle made by the tangent at $\theta = \pi / 3$ on the curve $x = a(\theta + \sin \theta)$,$y = a(1 - \cos \theta)$ with the $X$-axis is

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

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