If $x = \sqrt{2^{\csc^{-1} t}}$ and $y = \sqrt{2^{\sec^{-1} t}}$ for $|t| \ge 1$,then $\frac{dy}{dx}$ is equal to:

If $x = \sin t \cos 2t$ and $y = \cos t \sin 2t$,then at $t = \frac{\pi}{4}$,the value of $\frac{dy}{dx}$ is equal to

If $x=a \cos ^3 \theta$ and $y=a \sin ^3 \theta$,then $\frac{d^2 y}{d x^2}$ at $\theta=\frac{\pi}{4}$ is

At any two points of the curve represented parametrically by $x = a(2 \cos t - \cos 2t)$ and $y = a(2 \sin t - \sin 2t)$,the tangents are parallel to the $x$-axis. The values of the parameter $t$ corresponding to these points differ from each other by:

Find the derivative of $f(\sin x)$ with respect to $g(\sec x)$ at $x = \frac{\pi}{4}$, given that $f'(\frac{1}{\sqrt{2}}) = 3$ and $g'(\sqrt{2}) = 1$.

If $t$ is a parameter and $x=t+\frac{1}{t}, y=t-\frac{1}{t}$, then $\frac{d^2 y}{d x^2}=$