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 $f\left(x\right)=\left|\ln x\right|, x\ne 1$

What is the derivative of $f\left(x\right)$ at $x=0.5$?

Let $f$ and $g$ be two functions defined by $f\left(x\right)=\left\{\begin{array}{l}x+1, x<0\\ \left|x-1\right|, x\ge 0\end{array}\right.$and $g\left(x\right)=\left\{\begin{array}{l}x+1, x<0\\ 1, x\ge 0\end{array}\right.$.Then $\left(gof\right)\left(x\right)$ is

Let $\left[x\right]$ denote the greatest integer function and $f\left(x\right)=\max \left\{1+x+\left[x\right], 2+x, x+2\left[x\right]\right\}, 0\le x\le 2$, where $f$ is not continuous and $n$ be the number of points in $\left(0, 2\right)$, where $f$ is not differentiable. Then ${\left(m+n\right)}^2+2$ is equal to 

Consider the function $f\left(x\right)=\left[x^2-x\right]+\left\{x\right\}$, where, $\left[·\right]\to \mathrm{G}.\mathrm{I}.\mathrm{F}$ and $\left\{·\right\}$ is fractional part function.

If $f\left(x\right)=\left\{\begin{array}{l}3x^2+k\sqrt{x+1}; 0<x<1\\ 3m{x}^2+k^2; x\ge 1\end{array}\right.$ is differentiable at $x=1$, then $\frac{8f^{\text{'}}\left(8\right)}{f^{\text{'}}\left({\displaystyle \frac{1}{8}}\right)}$ for $k\ne 0$ is

If $f\left(x\right)=\left[x\right]\left[\mathrm{sin\pi }x\right]; x\in \left(-1,1\right)$, then $f\left(x\right)$ is (where, $\left[·\right]$ is GIF)

If $f\left(x\right)=\left(x^5+1\right)\left|x^2-4x-5\right|+\sin \left|x\right|+\cos \left|x-1\right|$, then $f\left(x\right)$ is not differentiable at 

Let $f\left(x\right)=\left\{\begin{array}{l}1-\sqrt{1-x^2}, \text{if} -1\le x\le 1\\ 1+\log \frac{1}{x}, \text{if} x>1\end{array}\right.$ is 

If $f\left(x\right)=x\left|x^2-3x\right|e^{\left|x-3\right|}+\left[\frac{3+\mathrm{sgn}\left(x^2-5x-1\right)}{4+x^2}\right]+\left\{\frac{1}{3}+\left[\frac{x^2-2x}{\left|x\right|+1}\right]\right\}$ is non-derivable at $x=x_1, x_2, x_3,....,x_n$, then $\sum _{i=1}^n{x}_i$ equals

Note: $\left\{·\right\}, \left[·\right], \left|·\right|$ represents fractional part, greatest integer function and modulus function respectively.

Let $f\left(x\right)=\max \left\{4,1+x^2,x^2-1\right\}$ $\forall x\in R$. Total number of points where $f\left(x\right)$ is non-differentiable is

If $f\left(x\right)=\left\{\begin{array}{ll}3,& x<0\\ 2x+1,& x⩾0\end{array}\right.$, then