If $x\sqrt{1+y}+y\sqrt{1+x}=0, x\neq y$, then $\frac{dy}{dx}$ is equal to

If $y = \log_x(x^2+1).\log_{(x^2+1)} x$, then $\frac{dy}{dx}$ is

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:

Points of discontinuity of the greatest integer function $f(x) = [x]$, where $[x]$ denotes integer less than or equal to $x$, are

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

The derivative of $|x|$ with respect to $x$, where $x \in R \sim \{0\}$ is

If $x = e^{\cos 2t}$, $y = e^{\sin 2t}$, then $\frac{dy}{dx}$ equals to