If $\sin x = \frac{2t}{1+t^2}$ and $\tan y = \frac{2t}{1-t^2}$,then $\frac{dy}{dx}$ is equal to

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) = $

If $x=\sec \theta-\cos \theta$ and $y=\sec ^n \theta-\cos ^n \theta$,then $\frac{d y}{d x}=$

What is the slope of the tangent to the curve $x = t^2 + 3t - 8$ and $y = 2t^2 - 2t - 5$ at the point $(2, -1)$?

Let $y = t^{10} + 1$ and $x = t^8 + 1,$ then $\frac{d^2y}{dx^2}$ is

The second order derivative of $a \sin^3 t$ with respect to $a \cos^3 t$ at $t = \frac{\pi}{4}$ is