If $f^{\prime}(x)=\sin (\log x)$ and $y=f\left(\frac{2 x+3}{3-2 x}\right)$,then $\frac{d y}{d x}$ at $x=1$ is

If $\sin y = \sin 3t$ and $x = \sin t$, then $\frac{dy}{dx} = $

Consider the curve represented parametrically by the equations $x = t^3 - 4t^2 - 3t$ and $y = 2t^2 + 3t - 5$,where $t \in \mathbb{R}$. If $H$ denotes the number of points on the curve where the tangent is horizontal and $V$ denotes the number of points where the tangent is vertical,then:

What is the slope of the normal to the curve $x = a(\theta - \sin \theta)$,$y = a(1 - \cos \theta)$ at the point $\theta = \pi / 2$?

If $x = 3 \tan t$ and $y = 3 \sec t,$ then the value of $\frac{d^2y}{dx^2}$ at $t = \frac{\pi}{4}$ is

If $x = a(1 - \cos \theta)$ and $y = a(\theta + \sin \theta)$,then $\frac{dy}{dx} = $ . . . . . . .