First Principle of Differentiation

Q. If $0<a,b<1$, and ${\tan }^{-1}a+{\tan }^{-1}b=\frac{\pi }{4}$, then the value of $(a+b)-\left(\frac{a^2+b^2}{2}\right)+\left(\frac{a^3+b^3}{3}\right)-\left(\frac{a^4+b^4}{4}\right)+....$

1. $e$
2. $e^2-1$
3. ${\log }_e\left(\frac{e}{2}\right)$
4. ${\log }_{e}2$

Q. Which of the following functions is/are continuous everywhere?

1. $\sin x$
2. $\tan x$
3. $|x|$
4. $a^x,\text{where}a>0$
5. Polynomial function

Q. If $f:R\to R$ satisfies $|f\left(x\right)-f\left(y\right)|\le |x-y{|}^3$ and $g\left(x\right)=f(-x+f\left(x\right)),$ then the value of $g^{\prime }\left(1\right)$ is

1. Cannot be determined
2. $1$
3. $-1$
4. $0$

Q. Total number of points of non-differentiability of $f\left(x\right)=[3+4sinx]$ in $[\pi ,2\pi ]$ where [.] denote the g.i.f are

1. 5
2. 6
3. 8
4. 9