Differentiability

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

A twice differentiable function $f\left(x\right)$ is defined for all real numbers and satisfies the following conditions, $f\left(0\right)=2; f^{\text{'}}\left(0\right)=-5 \text{and} f^{\text{'}\text{'}}\left(0\right)=3.$ The function $g\left(x\right)$ is defined by $$$g\left(x\right)=e^{ax}+f\left(x\right)\forall x\in R$, where $a$ is any constant. If $g^{\text{'}}\left(0\right)+g^{"}\left(0\right)=0.$ Then $a$ can be equal to 

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)$=\{\begin{array}{cc}{\text{tan}}^{-1}x& \text{if }\left|x\right|\le 1\\ \frac{1}{2}\left(\left|x\right|-1\right)& \text{if }\left|x\right|>1\end{array}\text{is}$

$f\left(x\right)=1+x.\left[\cos x\right]; 0<x\le \frac{\pi }{2}$, where $\left[\right]$ denotes greatest integer function then,

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 $x$?

The number of points at which the function $f\left(x\right)=max\left\{a-x, a+x, b\right\}, -\infty <x<\infty \text{, }0<a<\text{b}$ can not be differentiable is

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 $f\left(g\left(x\right)\right)$.

On differentiating  $\sqrt{\tan x}$ with respect to $x$ from first principle, the solution would be:

Find the set of points where $f\left(x\right)=x\left|x\right|$ is differentiable

Find the slope of the tangent at a point $\left(7,78\right)$ to the curve $y=x^2+4x+1$.

Find the slope of the tangent at a point $\left(-5,-6\right)$ to the curve $y=x^5+2x^2+1$.

Find the slope of the tangent at a point $\left(1,2\right)$ to the curve $y=4x^3$.

Find the slope of the tangent at a point $\left(5,6\right)$ to the curve $y=x^4+2x^2+x$.

Find the slope of the tangent at a point $\left(3,4\right)$ to the curve $y=x^3+3x^2+1$.

If $f\left(x\right)=x^2+qx+r, q^2-4r>0, q<0$ and $r>0$, then number of points where function $\left|f\left(\left|x\right|\right)\right|$ is not differentiable.