The differential coefficient of $\sin^{-1}x$ with respect to $\cos^{-1}\sqrt{1-x^2}$ is

If $x$ and $y$ are connected parametrically by the equations,without eliminating the parameter,find $\frac{dy}{dx}$ for $x = a(\cos \theta + \theta \sin \theta)$ and $y = a(\sin \theta - \theta \cos \theta)$.

The angle made by the tangent of the curve $x = a(t + \sin t \cos t)$; $y = a(1 + \sin t)^2$ with the $x$-axis at any point on it is

If $y = t^2 + t^3$ and $x = t - t^4$,then the value of $\frac{d^2 y}{d x^2}$ at $t = 1$ is equal to

For $x \neq -1, y \neq -1$, if $x = \frac{1 - \sqrt[3]{y}}{1 + \sqrt[3]{y}}$, then $\frac{dx}{dy} =$

For a positive constant $a$,find $\frac{dy}{dx}$,where $y = a^{t+\frac{1}{t}}$ and $x = \left(t+\frac{1}{t}\right)^{a}$.