Derivative from First Principle

Q.

Find the derivative of the following functions from first principle.

(i) $x^3$ - 27
(ii) (x-1) (x-2)

(iii) $\frac{1}{x^2}$
(iv) $\frac{x+1}{x-1}$

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

Q.

Find the derivative of:

(i) 2x - $\frac{3}{4}$
(ii) $(5x^3+3x-1)$ (x-1)

(iii) $x^{-3}$ (5+3x)
(iv) $x^5(3-6x^{-9})$

(v) $x^{-4}(3-4x^{-5})$
(vi) $\frac{2}{x+1}-\frac{x^2}{3x-1}$

Q.

Find the derivative of $\frac{x^n-a^n}{x-a}$ for some constant a.

Q. Find the derivative of x 2 – 2 at x = 10.

Q.

Select the correct statement fiven option for the function $f\left(x\right)=x{e}^{1-x}$

1. Strictly increases in the interval $\left(\frac{1}{2},2\right)$

2. Increases in the interval $\left(0,\infty \right)$

3. Decreases in the interval $\left(0,2\right)$

4. Strictly decreases in the interval $(1,\infty )$

Q. Find the derivative of 99 x at x = 100.

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)

Q. Find the derivative of x at x = 1.

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 the following function:
f(x)= $\frac{p{x}^2+qx+r}{ax+b}$

Q.

If ${lim}_{x\to a}f\left(x\right)=Land{lim}_{x\to a}g\left(x\right)=m$, then ${lim}_{x\to a}\left(f\right(x){)}^{g\left(x\right)}=L^m$

1. False

2. True

Q. Find the derivative of the following function:
f(x)= $\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$

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. 15. cosec x cot x

Q. The derivative of $f\left(x\right)=x^2$ is .

1. x
2. 2x
3. $x^3$
4. $x^2+1$

Q. Find the derivative of the following:
$(5x^3+3x-1)(x-1)$.

Q. Find the derivative of the following functions:
$5\sec x+4\cos x$

Q.

If y = $ta{n}^{-1}\left(\frac{1}{1+x+x^2}\right)+ta{n}^{-1}\left(\frac{1}{x^2+3x+3}\right)+ta{n}^{-1}\left(\frac{1}{x^2+5x+7}\right)$ + ........+

upto n terms they y (o) is equal to

1. 

2. 

3. 

4. 1+n

Q.

Find the derivative of $\frac{2x+3}{3x+2}$ by first principle.

Q. Find the derivative of
$\frac{2}{x+1}-\frac{x^2}{3x-1}$

Q. Find the derivative of the following functions (it is to be understood that a , b , c , d , p , q, r and s are fixed non-zero constants and m and n are integers):

Q. 29. (x +secx) (x -tanx)

Q.

The Derivative of $e^x$

1. $e^x$

2. $e^{x+1}$

3. $x{e}^x$

4. $e^{2x}$

Q.

Find the derivative of x at x = 1.

Q. 12. (ax +b)"

Q.

Find the derivative of $x^2$ - 2 at x = 10.

Q. The co-efficient of $x^{53}$ in the expansion $\sum _{m=0}^{100}{}^{100}{C}_m{\left(x-3\right)}^{100-m}{2}^m$ is

1. ${}^{-100}{C}_{47}$
2. ${-}^{100}{C}_{53}$
3. ${}^{65}{C}_{53}$
4. ${}^{100}{C}_{65}$

Q. 14. sin (r + a)