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

A function $f(x) = \begin{cases} \frac{kx^3}{4} & \text{if } x \leq 1 \\ 8 & \text{if } x > 1 \end{cases}$ for some constant $k$, then $\frac{d^2f(x)}{dx^2}$ at $x = -1$ is

For the function $f(x) = \sqrt{x}$, which of the following statements are true?

(A) $f(x)$ is continuous for $x \geq 0$.

(B) $f(x)$ is continuous for $x \in R$ (Set of real numbers).

(C) $f(x)$ is differentiable for $x > 0$.

(D) $f(x)$ is differentiable for $x \in R$ (Set of real numbers).

Choose the correct answer from the options given below:

The graph of a function $f: R \to R$ is shown below, where $R$ is a real numbers

Then which of following statements is correct?

If $x = 3t^2$ and $y = \sqrt{t}$, then the value of $\frac{d^2y}{dx^2}$ is

If $y = \log\left[\frac{x^2}{e^2}\right]$ then value of $\frac{d^2y}{dx^2}$ is :

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