Let a function $f(x)=\alpha+(\beta^2+5\beta+6)|x|+\gamma|x|^4$, where $\alpha, \beta$ and $\gamma$ are real number. Then the function $f(x)$ is differentiable at $x=0$ if

If $y = \left(\log\left(x + \sqrt{x^2+a^2}\right)\right)^2$ and $x \neq \frac{1-a^2}{2}$, then $(x^2+a^2)\frac{d^2y}{dx^2} + x\frac{dy}{dx}$ is equal to:

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

If $f$ is a function defined by $f(x) = \begin{cases} 5x^2 - x + 3, & x < 1 \\ 3x + 4, & x \geq 1 \end{cases}$, then, at $x = 1$, $f$ is

If $f(x)=\begin{cases}\frac{\sin(a-b)x}{x}, & x<0\\ 2a+b, & x=0\\ \frac{-\tan 12x}{bx}, & x>0\end{cases}$, $(b>0)$ is continuous at $x=0$, then

Let $f(x)=\begin{cases}\dfrac{|x|}{x},&x\ne0\\1,&x=0\end{cases}$ and $g(x)=\begin{cases}x\sin\left(\dfrac{1}{x}\right),&x\ne0\\0,&x=0\end{cases}$

Then at the origin, which one of the following is true?

If $x = 4t$ and $y = \frac{4}{t}$, then $\frac{d^2y}{dx^2}$ is