If $t=e^{2 x}$ and $y=\log _{e} t^{2}$, then $\frac{d^{2} y}{d x^{2}}$ is :

If $f(x) = \begin{cases} \dfrac{1}{|x|}, & x \geq 1 \\ ax^2+b, & x < 1 \end{cases}$ is differentiable at $x=1$ then

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

If $x=2at^5$ and $y=at^{10}$, then $\dfrac{dy}{dx}=$

Let $y = \log(x + \sqrt{x^2+1})$, and $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} = c$. Then identify the correct statements about the values of $a$, $b$ and $c$ :

(A) $a = 1 + x^2$

(B) $b = 0$

(C) $c = 0$

(D) $b = x$

(E) $c = 2$

Choose the correct answer from the options given below :

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 $y = \log(\log x)$, then $\frac{d^2y}{dx^2}$ is equal to: (Consider $\log_e x = \log x$)