Derivative from First Principle

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

1. $f^{\prime }$ does not exist at $x=0$
2. $f^{\prime }$ exists and is continuous at $x=0$
3. $f^{\prime }$ exists but not continuous at $x=0$
4. $f^{\prime }$ does not exist at any point

Q.

The Derivative of $e^x$

1. $e^x$

2. $e^{x+1}$

3. $x{e}^x$

4. $e^{2x}$

Q. Two functions $f:R\to R,g:R\to R$ are defined as follows $f\left(x\right)=\{\begin{array}{cc}0,& x\in Q\\ 1,& x\notin Q\end{array}$, $g\left(x\right)=\{\begin{array}{cc}-1& x\in Q\\ 0,& x\notin Q\end{array}$, then $g\left(f\right(e\left)\right)+f\left(g\right(\pi \left)\right)=$

1. $2$
2. $-1$
3. $0$
4. $1$

Q.

Find the Derivative of Sin x.

1. sin x

2. cos x

3. -cos x

4. -sin x

Q.

Find the Derivation of $f\left(x\right)=x^2.atx=0$

1. 0

2. 1

3. -1

4. 2