The derivative of $\sec(\tan \sqrt{x})$ with respect to x is :

If

$f(x) = \begin{cases} mx + 1,\ x \geq \pi/2 \\sin x + n, x \leq \pi/2, & \end{cases}$

is continuous at $x = \pi/2$, where $m \in \mathbb{Z}$ (set of integers), then $\sin 2n =$

If $e^y(x + 1) = 1$ and $\frac{d^2y}{dx^2} = k(\frac{dy}{dx})^2$, then k is equal to

If a function $f(x) = \begin{cases}\lambda(x^2-2x), & x \le 0\\4x+1, & x>0\end{cases}$ then

If $x = 3\sin\theta - \sin 3\theta$, $y = 3\cos\theta - \cos 3\theta$, then $\frac{d^2y}{dx^2}$ at $\theta = \frac{\pi}{3}$ is

Let $[x]$ denote the greatest integer $\leq t$ and $a \mathbb{Z} = [ax: x \in \mathbb{Z}, a \in \mathbb{R}]$ (where $\mathbb{Z}$ is set of integer and $\mathbb{R}$ is set of real number). The set of points of discontinuity of the function $f(x) = [2x]$ is given by

If $y = x^{\sin x}$, then value of $x^{-\sin x} \frac{dy}{dx} - \cos x \log x$ is :