Differentiability

$\text{Let\hspace{0.17em}}f\left(x\right)=\{\begin{array}{l}\frac{\sin x}{x}x\ne 0\\ 1x=0\end{array}\text{\hspace{0.17em}find\hspace{0.17em}}{f}^{\text{'}\text{'}}\left(0\right)\left(\text{if\hspace{0.17em}}\text{\hspace{0.17em}it\hspace{0.17em}}\text{\hspace{0.17em}exists}\right).$

Let g(x) be a polynomial of degree one & f(x) be defined by $\text{f}\left(x\right)=[\begin{array}{cc}\text{g}\left(x\right)\text{,}& x\le 0\\ {\left(\frac{1+x}{2+x}\right)}^{1/x}\text{,}& x>0\end{array}$ such that  f(x) is continuous $f^{\text{'}}\left(1\right)=f\left(-1\right)$, then g(x) is

The domain of the derivative of the function f(x)$=\{\begin{array}{cc}{\text{tan}}^{-1}x& \text{if }\left|x\right|\le 1\\ \frac{1}{2}\left(\left|x\right|-1\right)& \text{if }\left|x\right|>1\end{array}\text{is}$

Consider the function $f\left(x\right)=x^2-2x$ and $g\left(x\right)=-\left|x\right|$ 
Statement-1: The composite function $F\left(x\right)=f\left(g\left(x\right)\right)$ is not derivable at $x=0$.
Statement-2: $F\text{'}\left(0^{+}\right)=2$ and $F\text{'}\left(0^{-}\right)=-2.$

Let $f\left(x\right)=[n+p\sin x],x\in (0,\pi ),n\in I$and$p$ is a prime number. The number of points where $f\left(x\right)$ is not differentiable is

( Here $\left[x\right]$ represents the greatest integer less than or equal to $x$ )

The set of all points where the function $\text{f}\left(x\right)=\frac{x}{1+\left|x\right|}$ is differentiable is:

$Givenf\left(x\right)=[\begin{array}{c}{\log }_a{\left(a|\left[x\right]+\left[-x\right]|\right)}^x\left(\frac{\left(a\frac{2}{\left(\frac{\left[x\right]+\left[-x\right]}{\left|x\right|}\right)}\right)-5}{3+\frac{1}{\left|x\right|}}\right)for\left|x\right|\ne 0;a>1\\ 0forx=0\end{array}$ where [ ] represent

integral part function, then:

For what triplets of real numbers $\left(a, b ,c\right)$ with $\mathrm{a}\ne 0$ the function $f\left(x\right)=\left\{\begin{array}{ll}x,& x\le 1\\ a{x}^2+bx+c,& otherwise\end{array}\right.$is differentiable for all $x$?

If $f\left(x\right)=\left|\sin x\right|$ and $g\left(x\right)=x^3$, then identify which of the following is correct for the function $f\left(g\left(x\right)\right)$.

Let $f\left(x\right)=\left\{\begin{array}{ll}{x}^2\sin \frac{1}{x},& x\ne 0\\ 0,& x=0\end{array}\right.$. Then 

If the function $f:R\to R$, defined by $f\left(x\right)=\left\{\begin{array}{cc}ax,& x<2\\ a{x}^2-bx+3,& x\ge 2\end{array}\right.$  is differentiable, then the value of $f^{\text{'}}\left(-3\right)+f^{\text{'}}\left(3\right)$ is equal to

Let $f\left(x\right)=\left\{\begin{array}{cl}\frac{1}{\left|x\right|},& \left|x\right|>2\\ a+b{x}^2,& \left|x\right|\le 2\end{array}\right.$, then $f\left(x\right)$ is differentiable at $x=-2$ for

The function$f\left(x\right)=\left|2\mathrm{sgn}\right(2x\left)\right|+2$ is (where $\mathrm{sgn}\left(x\right)$ means signum $x$)

The function $f\left(x\right)=\left|x\right|$ at $x=0$ is

If $f\left(x\right)=p\left|\sin x\right|+q{e}^{\left|x\right|}+r{\left|x\right|}^3$ and if $f\left(x\right)$ is differentiable at $x=0$, then

The number of points at which the function $f\left(x\right)=\left|x-0.5\right|+\left|x-1\right|+\tan x$ is not differentiable in the interval $\left(0,2\right)$ is/are

Let $\mathrm{ƒ}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\left[\mathrm{x}\right]& \mathrm{x}\notin \mathrm{I}\\ \mathrm{x}-1& \mathrm{x}\in \mathrm{I}\end{array}\right.$ (where, $\left[ \right]$ denotes the greatest integer function) and

$\mathrm{g}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\sin \mathrm{x}+\cos \mathrm{x},& \mathrm{x}<0\\ 1,& \mathrm{x}\ge 0\end{array}\right.$ . Then 

If $f\left(x+y\right)=f\left(x\right)·f\left(y\right), \forall x, y\in R$ and $f\left(3\right)=2,$ $f\text{'}\left(0\right)=11, f\left(0\right)\ne 0$ then $f\text{'}\left(3\right)=$

If $f\left(x+y\right)=f\left(x\right)+f\left(y\right), \forall x, y\in R$ and $f\left(2\right)=5$ and $f\text{'}\left(0\right)=1,$ then the value of $f\text{'}\left(2\right)$ is

If $f\left(x\right)={\left|x\right|}^3$ then $f\text{'}\left(0\right)=$