If $x^2 - y^2 = 1$, then which of the following is correct? (A) $(x^2 - 1)\left(\frac{dy}{dx}\right)^2 = x^2$ (B) $(x^2 - 1)\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (C) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = x^2$ (D) $(x^2 - 1)^3\left(\frac{d^2y}{dx^2}\right)^2 = 1$ Choose the correct answer from the options given below:

The value of derivative of the function $\cot^{-1}\{(\cos 2x)^{1/2}\}$ at $x = \frac{\pi}{6}$ is

If $y = \left(x + \sqrt{x^2+1}\right)^m$, then $\frac{dy}{dx}$ is

If the function defined by $f(x) = \begin{cases} \ kx^2 + 1, & \text{if } x \le 1 \\ 2 , & \text{if } x > 1 \end{cases}$ is continuous at $x = 1$, then k is equal to

For the function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = \lvert x-3 \rvert + \lvert x+4 \rvert$, Where $\mathbb{R}$ is set of real numbers, which of the following statements are correct?

(A) $f(x)$ is continuous as well as differentiable at $x=3$.

(B) $f(x)$ is continuous but not differentiable at $x=3$ and $x=-4$

(C) $f(x)$ is neither continuous nor differentiable at $x=3$

(D) $f(x)$ is continuous on $\mathbb{R}$

Choose the correct answer from the options given below:

If $y = \log_e \sin(e^x + 5x + 500)$ then $\dfrac{dy}{dx}$ is equal to

If $x=\log_e t$ and $y=\dfrac{1}{t}$, then $\dfrac{d^2y}{dx^2}$ at $t=0.1$ is