If $y = \tan^{-1} \sqrt{\frac{1 + \sin 2x}{1 - \sin 2x}}$, then at $x = \frac{\pi}{6}$, the value of $\frac{dy}{dx}$ is

Let $P$ be a non-zero polynomial such that $P(1+x)=P(1-x)$ for all real $x$ and $P(1)=0$. Let $m$ be the largest integer such that $(x-1)^m$ divides $P(x)$ for all such $P(x)$. Then,$m$ equals

The derivative of $\tan x - x$ with respect to $x$ is

If $y=\frac{\cos x}{1+\sin x}$,then
$(a)$ $\frac{dy}{dx}=\frac{-1}{1+\sin x}$
$(b)$ $\frac{dy}{dx}=\frac{1}{1+\sin x}$
$(c)$ $\frac{dy}{dx}=-\frac{1}{2} \sec ^2\left(\frac{\pi}{4}-\frac{x}{2}\right)$
$(d)$ $\frac{dy}{dx}=\frac{1}{2} \sec ^2\left(\frac{\pi}{4}-\frac{x}{2}\right)$

If $2f(\sin x) + f(\cos x) = x,$ then $\frac{d}{dx} f(x)$ is

Find the derivative of the function $(ax+b)(cx+d)^{2}$ with respect to $x$,where $a, b, c, d$ are fixed non-zero constants.