Match the LIST-I with LIST-II

LIST-I	LIST-II
A. $f(x) = \log_e(x^2+1)$	I. not continuous at all integral values
B. $f(x) = \lvert x\rvert$	II. is continuous as well as differentiable at $x=0$
C. $f(x) = x^2$	III. Continuous everywhere
D. $f(x) = [x]$, $[x]$ is greatest integer function	IV. Continuous everywhere but not differentiable at $x=0$

Choose the correct answer from the options given below:

If $e^x + e^y = e^{x+y}$, then $\frac{dy}{dx}$ equals

The derivative of $(\log x)^x$ with respect to $\log x$ is

If $f(x) = \begin{cases} \frac{\tan(\pi/4 - x)}{\cot 2x}, & x \neq \pi/4 \\ k, & x = \pi/4 \end{cases}$ is continuous at $x = \pi/4$, then the value of k is

Match List-I with List-II

Choose the correct answer from the options given below:

If $y = \sin^{-1} x + \sin^{-1} \sqrt{1-x^2}, x \in (-1, 0)$, then $\frac{dy}{dx}$ is equal to

If $y = \dfrac{\log_e x}{x}$, then $\dfrac{d^2y}{dx^2}$ is :