First Principle of Differentiation

Q. Let $f:R\to R$ satisfying $|f\left(x\right)|\le x^2\forall x\in R$ is differentiable at $x=0,$ then $f^{\prime }\left(0\right)$ is

Q.

What is the derivative at the point $x=-2$ on the curve $y=x^2$?
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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. 9
4. 8

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. Let $f\left(x\right)=\underset{h\to 0}{lim}\frac{\left(\sin \right(x+h){)}^{\ln (x+h)}-(\sin x{)}^{\ln x}}{h}$ then $f\left(\frac{\pi }{2}\right)$ is

1. Equal to $0$
2. Equal to $1$
3. $\ln \frac{\pi }{2}$
4. Non - existent

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. Let $f\left(x\right)=\{\begin{array}{c}\frac{x}{1+e^{1/x}}x\ne 0,\\ 0x=0\end{array}.$ Then

1. $f^{\prime }\left(0^{-}\right)=0$
2. $f^{\prime }\left(0^{-}\right)=1$
3. $f^{\prime }\left(0^{+}\right)=0$
4. $f^{\prime }\left(0^{+}\right)=1$

Q. Suppose $p\left(x\right)=a_0+a_{1}x+\cdots +a_n{x}_n$. If $|p\left(x\right)|\le |e^{x-1}-1|$ for all $x\ge 0$, then the maximum value of $|a_1+2a_2+\cdots +n{a}_n|$ is equal to

Q. If $f\left(x\right)=|x^2-5x+6|$, then $f^{\prime }\left(x\right)$ equals

1. $2x-5$ for $2<x<3$
2. $5-2x$ for $2<x<3$
3. $2x-5$ for $x<2$
4. $5-2x$ for $x<3$

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