Differentiation from 1st Principles

Q.

Find the derivative of $\frac{1}{\sqrt{x}}$

Q.

What is the derivative of $\frac{1}{x}$?

Q. Find the derivative of functions $cosx$, $cosecx$, $secx$, $cotx$ using first principle​​​​​​

1. $sinx,-cosecxcotx,secxtanx,-cose{c}^{2}x$
2. $-sinx,cosecxcotx,-secxtanx,cose{c}^{2}x$
3. $-sinx,-cosecxcotx,secxtanx,-cose{c}^{2}x$
4. $-sinx,-cosecxcotx,-secxtanx,cose{c}^{2}x$

Q. Differentiation is basically a combination of and .

1. summation
2. difference
3. multiplication
4. division

Q. If the displacement of particle at any time $t$ is given by $s=ut+\frac{1}{2}a{t}^2$ where $u$ is initial velocity (constant) and $a$ is acceleration (constant) then, velocity of particle at any time $t$ will be

1. $u+2at$
2. $u+\frac{at}{2}$
3. $\frac{u{t}^2}{2}+\frac{a{t}^3}{6}$
4. $u+at$

Q. A bus moving with uniform acceleration of $2m/s^2$ passes with speed 5 m/s when a car acceleration at $4m/s^2$ overtakes it. the velocity of the car is found to be 50 /m after 10s. the distance of separation between them after 10 s is

1. $150m$
2. $300m$
3. $75m$
4. $250m$

Q. If the displacement of a particle at any time $t$ is given by $s=ut+\frac{1}{2}a{t}^2$, where $u$ and $a$ are constants. Then the acceleration of particle at any time $t$ will be

1. $u+a$
2. $u$
3. None
4. $a$

Q. Consider the following pairs of quantities:
1. Young's modulus; pressure
2. Torque; energy
3. Linear momentum; work
4. Solar day; light year
In which cases are the dimensions, within a pair, same?

1. 1 and 4
2. 2 and 4
3. 1 and 2
4. 1 and 3

Q. The area $A$ of a circle is related to its diameter by the equation $A=\frac{\pi }{4}{D}^2$. The change in area with respect to the diameter is $x\times \pi$, when the diameter is $10m$. Find the value of $x$.

Q.

What is the derivative of $x^x$?

Q. A particle experiences constant acceleration for $6s$ after starting from rest. If it travels a distance $s_1$ in the first $2s$, a distance $s_2$ in the next $2s$ and a distance $s_3$ in the last $2s$, then $s_1:s_2:s_3$ is:

1. $1:1:1$
2. $1:2:3$
3. $1:3:5$
4. $1:5:9$

Q.

The displacement of the particle (in $m$) is given by

$y=2t+t^2-2t^3$

What is the velocity (in $m/s$) of the particle at the instant when its acceleration is zero?

1. $\frac{13}{6}$
2. $\frac{6}{13}$
3. $\frac{5}{2}$
4. $\frac{2}{5}$

Q. If the displacement of a particle at any time $t$ is given by $s=ut+\frac{1}{2}a{t}^2$, where $u$ and $a$ are constants. Then the acceleration of particle at any time $t$ will be

1. $a$
2. $u+a$
3. $u$
4. None

Q. Using the concept of limit, slope of a point can be calculated.

1. True
2. False

Q. Find the derivative of $x^2$ by first principle

Q. Find the derivative of $x^2$ using first principle.

1. $2x^2$
2. $\frac{x}{2}$
3. $2x$
4. $\frac{x^2}{2}$

Q. Find the derivative of $x^2$ using first principle.

1. $2x^2$
2. $\frac{x}{2}$
3. $2x$
4. $\frac{x^2}{2}$

Q. If the displacement of a particle at any time $t$ is given by $s=ut+\frac{1}{2}a{t}^2$, where $u$ and $a$ are constants. Then the acceleration of particle at any time $t$ will be

1. $a$
2. $u+a$
3. $u$
4. None

Q.

A curve is represented by y=sin x. If x is changed from $\frac{\pi }{3}to\frac{\pi }{3}+\frac{\pi }{100}$, find approximately the change in y.

1. $\frac{\pi }{100}$

2. $\frac{\pi }{200}$

3. $\frac{1}{2}$

4. None of these

Q. The horizontal distance between two bodies, when their velocity are perpendicular to each other, is

1. $1m$
2. $0.5m$
3. $4m$
4. $2m$

Q.

A curve is represented by y=sin x. If x is changed from $\frac{\pi }{3}to\frac{\pi }{3}+\frac{\pi }{100}$, find approximately the change in y.

1. 

2. 

3. 

4. None of these

Q. Find the derivative of $x^2$ using first principle.

1. $2x^2$
2. $\frac{x}{2}$
3. $2x$
4. $\frac{x^2}{2}$

Q. The derivative of a constant is some constant.

1. False
2. True

Q. The area $A$ of a circle is related to its diameter by the equation $A=\frac{\pi }{4}{D}^2$. The change in area with respect to the diameter is $x\times \pi$, when the diameter is $10m$. Find the value of $x$.

Q. The position $x$ of a particle varies with time $\left(t\right)$ as $x=a{t}^2-b{t}^3$. The acceleration at time $t$ of the particle will be equal to zero, where $t$ is equal to

1. $\frac{2a}{3b}$
2. $\frac{a}{b}$
3. zero
4. $\frac{a}{3b}$

Q. Differentiation is basically a combination of and .

1. summation
2. difference
3. multiplication
4. division

Q. Which of the following changes when a particle is moving with uniform velocity?

1. Speed
2. Velocity
3. Acceleration
4. Position vector

Q. Using the concept of limit, slope of a point can be calculated.

1. True
2. False

Q. The area $A$ of a circle is related to its diameter by the equation $A=\frac{\pi }{4}{D}^2$. The change in area with respect to the diameter is $x\times \pi$, when the diameter is $10m$. Find the value of $x$.

Q. A body at some point of time is having $MOI$ as 2 units and angular velocity as 3 units, and their rates of change are -1 and 3 then, torque at that point of instant is