Derivative from First Principle

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

Q.

The Derivative of $e^x$

1. $e^{x+1}$

2. $x{e}^x$

3. $e^{2x}$

4. $e^x$

Q. Let $f:R\to R$ be a differentiable function satisfying $f(x+y)=f\left(x\right)+f\left(y\right)+xy$ for all $x,y\in R$ and $\underset{h\to 0}{lim}\frac{1}{h}f\left(h\right)=3.$ Which of the following statements is (are) CORRECT?

1. $f$ is a linear function
2. $2f\left(1\right)=\left[\underset{x\to 0}{lim}\right(1+2x{)}^{1/x}]$, where $[.]$ denotes the greatest integer function
3. The minimum value of $f$ is $-\frac{9}{2}$
4. $f^{\prime }\left(1\right)=4$

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. Let $f$ be any function defined on $R$ and let it satisfy the condition :
$|f\left(x\right)-f\left(y\right)|\le |(x-y{)}^2|,\forall x,y\in R$
If $f\left(0\right)=1,$ then

1. $f\left(x\right)$ can take any value in $R.$
2. $f\left(x\right)=0,\forall x\in R$
3. $f\left(x\right)>0,\forall x\in R$
4. $f\left(x\right)<0,\forall x\in R$

Q. Find the derivative of the following functions from first principle. (i) x 3 – 27 (ii) ( x – 1) ( x – 2) (ii) (iv)

Q. Find the derivative of cos x from first principle.

Q. Find the derivative of the following functions from first principle: (i) – x (ii) (– x ) –1 (iii) sin ( x + 1) (iv)

Q. 25. (x +cosx) (x- tanx)