If $y = \cot^{-1}(x^2)$,then $\frac{dy}{dx}$ is equal to

Find the derivative of the following function: $\operatorname{cosec} x \cot x$.

Find the derivative of the function: $(x^{2}+1) \cos x$. (Assume that $a, b, c, d, p, q, r, s$ are fixed non-zero constants and $m, n$ are integers.)

If $f(x)=3 x^2+2 x f^{\prime}(1)+f^{\prime \prime}(2)$,then $f(x)=$ . . . . . .

If $y = x \left[ \left( \cos \frac{x}{2} + \sin \frac{x}{2} \right) \left( \cos \frac{x}{2} - \sin \frac{x}{2} \right) + \sin x \right] + \frac{1}{2\sqrt{x}}$,then $\frac{dy}{dx} = $

$A$ function $f$ satisfying $f'( \sin x ) = \cos^2 x$ for all $x$ and $f(1) = 1$ is :