If $x = 2 \cos^3 \theta$ and $y = 3 \sin^2 \theta$,then $\frac{dy}{dx} = $

For what value of $t$ is the tangent to the curve $x = t^2 - 1, y = t^2 - t$ perpendicular to the $x$-axis?

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

If $x=a\left\{\cos \theta+\log \tan \left(\frac{\theta}{2}\right)\right\}$ and $y=a \sin \theta$, then $\frac{dy}{dx}$ is equal to

If $x=a(t-\sin t)$ and $y=a(1+\cos t)$, then $\frac{d^2 y}{d x^2}=$

If $x = a(\cos \theta + \theta \sin \theta)$, $y = f(\theta)$, $f(2\pi) = 0$, $\frac{dy}{dx} = \frac{\tan \theta}{\theta}$, $\theta \neq 0$ and $\theta \neq (2n+1)\frac{\pi}{2}$, then $f\left(\frac{\pi}{3}\right) = $