Let the function $y=f(x)$ be given by $x=t^{5}-5t^{3}-20t+7$ and $y=4t^{3}-3t^{2}-18t+3$, where $t\epsilon \left ( -2, 2 \right )$. Then $f^{'}(x)$ at $t=1$ is ?

If for $x \in \left(0, \dfrac{1}{4}\right)$, the derivative $\tan^{-1} \left(\dfrac{6x\sqrt{x}}{1-9x^{3}}\right)$ is $\sqrt{x}.g(x)$, then $g(x)$ equals :

If $x=3\sin {t} ,\ y=3\cos {t} ,$ find $\dfrac {dy}{dx}$ at $t=\dfrac { \pi }{ 3 }$

Let $g$ is the inverse function of $f$ and $f'(x)=\dfrac {x^{10}}{(1+x^{2})}$. If $g(2)=a$ then $g'(2)$ is equal to

If $f(x) = \sin^{-1} \left ( \frac{4^{x + \frac{1}{2}}}{1 + 2^{4x}} \right )$, which of the following is not the derivative of f(x)?