List of top Mathematics Questions on Derivatives of Functions in Parametric Forms asked in MHT CET

If $x = \tan^{-1} \left\{ \frac{\sqrt{1+t^2}-1}{t} \right\}, y = \cos^{-1} \left\{ \frac{1-t^2}{1+t^2} \right\}$, then $\frac{dy}{dx}$ is equal to

If \[ 3\sin^{-1}\left(\frac{2x}{1+x^2}\right) - 4\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) + 2\tan^{-1}\left(\frac{2x}{1-x^2}\right) = \frac{\pi}{3}, \] then the value of \[ x = \, ? \] 

If $x = \tan^{-1} \left\{ \frac{\sqrt{1+t^2}-1}{t} \right\}, y = \cos^{-1} \left\{ \frac{1-t^2}{1+t^2} \right\}$, then $\frac{dy}{dx}$ is equal to
If $x = \tan^{-1} \left\{ \frac{\sqrt{1+t^2}-1}{t} \right\}, y = \cos^{-1} \left\{ \frac{1-t^2}{1+t^2} \right\}$, then $\frac{dy}{dx}$ is equal to
If the lines \( \frac{x-k}{2} = \frac{y+1}{3} = \frac{z-1}{4} \) and \( \frac{x-3}{1} = \frac{y - 9/2}{2} = z/1 \) intersect, then the value of \( k \) is:
The derivative of $f(\sec x)$ with respect to $g(\tan x)$ at $x=\frac{\pi}{4}$, where $f^{\prime}(\sqrt{2})=4$ and $g^{\prime}(1)=2$, is
If $f'(x) = \sin(\log x)$ and $y = f\left(\frac{2x+3}{3-2x}\right)$, then $\frac{dy}{dx}$ at $x=1$ is
If $x = a(t + \sin t), y = a(1 - \cos t)$, then $\frac{dy}{dx} =$
If $x = a \cos \theta$, $y = b \sin \theta$, then $\left[\frac{d^2 y}{d x^2}\right] = $