If $y = ax^{n+2} + \dfrac{b}{x^{n+1}}$, then $x^2\dfrac{d^2y}{dx^2}$ is equal to

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

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

If $x = a(\theta+\sin\theta)$, $y = a(1-\cos\theta)$ then $\dfrac{d^2y}{dx^2}$ is equal to:

Let $e^y(x+1) = 1$. Then which of the following are TRUE?

(A) $\frac{d^2y}{dx^2} = -\frac{1}{(x+1)^2}$

(B) $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$

(C) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = -1$

(D) $\left.\frac{d^2y}{dx^2}\right|_{x=0} = 1$

(E) $\left.\frac{d^2y}{dx^2}\right|_{x=1} = \frac{1}{4}$

Choose the correct answer from the options given below:

Which of the following functions $f(x)$ are differentiable at $x = 0$?

(A) $|x|$

(B) $|x - 1|$

(C) $[x]$, where $[t]$ denotes the greatest integer $\leq t$

(D) $|x + 1|$

(E) $x^2$

Choose the correct answer from the options given below:

If $y = x^x$, then value of $\frac{dy}{dx}$ at $x = 2$ is :