Basic Differentiation Rule

Q.

How do you find the derivative of $\frac{1}{x^2}$?

Q.

$\frac{d^{2}x}{d{y}^2}$is equal to

1. ${\left(\frac{d^{2}y}{d{x}^2}\right)}^{-1}$

2. $-\left(\frac{d^{2}y}{d{x}^2}\right)-1{\left(\frac{dy}{dx}\right)}^{-3}$

3. $-\left(\frac{d^{2}y}{d{x}^2}\right){\left(\frac{dy}{dx}\right)}^{-3}$

4. $noneofthese$

Q.

What is the derivative of a function?

Q. If $y=\tan x.{\cos }^{2}x$ then $\frac{dy}{dx}$ will be

1. $1+2{\sin }^{2}x$
2. $1-2{\sin }^{2}x$
3. 1
4. $2{\sin }^{2}x$

Q. If $y=3t^2-4t$, then minima of $y$ will be at

1. $t=3/2$
2. $t=3/4$
3. $t=2/3$
4. $t=4/3$

Q. If $y=e^x.\cot x$ then $dy/dx$ will be

1. $e^x\cot x-cose{c}^{2}x$
2. $e^x({\cot }^{2}x-cose{c}^{2}x)$
3. $e^x\cot x-cosecx$
4. $e^x(\cot x-cose{c}^{2}x)$

Q. If $y=sin\left(x\right)+ln\left(x^2\right)+e^{2x}$, then $dy/dx$ is

1. $cosx+\frac{2}{x}+e^{2x}$
2. $cosx+\frac{2}{x}+2e^{2x}$
3. $-cosx+\frac{2}{x^2}+e^{2x}$
4. $-cosx-\frac{2}{x^2}+2e^{2x}$

Q. A boy throws a stone in the air, its height at any time is given by $h=-3t^2+12t-4\text{m}$. The maximum height (in $\text{m}$) attained by the stone is

1. $12\text{m}$
2. $20\text{m}$
3. $8\text{m}$
4. $16\text{m}$

Q. If velocity of particle is given by $v=2t^4$, then its acceleration $\frac{dv}{dt}$ at any time $t$ will be given by

1. $8t^3$
2. $8t$
3. $t^2$
4. $-8t^3$

Q. If $y=e^{2^x}-\cos \left(\ln x\right)$, then $\frac{dy}{dx}$ is

1. $e^{2^x}+\frac{\sin \left(\ln x\right)}{x}$
2. $2^x{e}^{2^x}\ln 2+\frac{\sin \left(\ln x\right)}{x}$
3. $e^{2^x}+\sin \left(\ln x\right)$
4. $2^x{e}^{2^x}\ln 2-\sin \left(\ln x\right)$

Q.

If y=ln⁡x , then $\frac{dy}{dx}$ will be

Q. If motion of a particle is represented by $s=e^t+4^t-\cos t$, then the acceleration of the particle at any time $t$ is
[Hint: Acceleration of the particle is $\frac{d^{2}s}{d{t}^2}$]

1. $e^t-4^t\ln 4+\cos t$
2. $e^t+4^t(\ln 4{)}^2+\cos t$
3. $e^t+4^t(\ln 4{)}^2-\cos t$
4. $e^t+4^t\ln 4+\cos t$

Q. If $y=x^3+2x^2+7x+8$, then $\frac{dy}{dx}$ will be

1. $3x^2+2x+15$
2. $3x^2+4x+7$
3. $x^3+2x^2+15$
4. $x^3+4x+7$

Q. Maximum value of $f\left(x\right)=\sin x+\cos x$ is

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

Q.

The electric current in a charging R-C circuit is given by i= $i_0{e}^{-1/RC}$ where $i_0$, R and C are constant parameters of the circuit and t is time. Find the rate of change of current at (a) $t=0$, (b) t = RC, (c) t = 10 RC.

Q. If charge flown through a wire is given by $q=3\sin \left(3t\right)$, then find out the current flown through the wire at $t=\frac{\pi }{9}$ seconds.

1. $4.5\text{A}$
2. $9\text{A}$
3. $4.5\sqrt{3}\text{A}$
4. $\sqrt{3}/2\text{A}$

Q. Find derivative of $f\left(x\right)=e^{\cos x}+\ln x$

1. $-\cos x.e^{\sin x}+\frac{1}{x}$
2. $-\sin x.e^{-\cos x}+\frac{1}{x}$
3. $\sin x.e^{\cos x}+\frac{1}{x}$
4. $-\sin x.e^{\cos x}+\frac{1}{x}$

Q.

Water pours out at the rate of Q from a tap, into a cylindrical vessel of radius r. find the rate at which the height of water level rises when the height is h.

1. 

2. 

3. 

4. none of these

Q. The position of a particle moving along x-axis varies with time $t$ as $x=4t-t^2+1$. If the time interval(s) during which the particle is moving along positive x-direction is $0<t<u$. The value of $u$ (in $s$) is

Q.

$x^{12}-y^{12}$ is equal to ?

Q. If sides of a rectangle with given perimeter are $a$ & $b$, then find the relation between $a$ & $b$ for which area of the given rectangle is maximum.

1. $a+b=0$
2. $a.b=1$
3. $a=4b$
4. $a=b$

Q. In the given circuit, when the switch is closed, the capacitor charges with a time constant of

1. $RC$
2. $2RC$
3. $\frac{1}{2}RC$
4. $RC\log 2$

Q. If motion of a particle is represented by $s=e^t+4t^3-\cos t$, then the acceleration of the particle at any time $t$ is
[Hint: Acceleration of the particle is $\frac{d^{2}s}{d{t}^2}$]

1. $e^t+12t^2+\sin t$
2. $e^t+24t-\cos t$
3. $e^t+24t+\cos t$
4. $e^t+12t^2-\sin t$

Q. Find derivative of $f\left(x\right)=\frac{\cos x}{1+\sin x}$
[Hint: ${\sin }^2\theta +{\cos }^2\theta =1$]

1. $\frac{1}{(1+\sin x)}$
2. $\frac{-1}{(1+\sin x)}$
3. $\frac{-1}{(1-\sin x)}$
4. $\frac{-1}{(1-\cos x)}$

Q.

Water is flowing into a vertical cylindrical tank of radius $2$ft. at the rate of $8$ cubic ft. per minute.

The water level is rising at the speed of (ft . per minute)

1. $\frac{\mathrm{\pi }}{2}$

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

3. $2$

4. None of these

Q.

A box with a square base and open top must have a volume of $4000c{m}^3$.

Find the dimensions of the box that minimize the amount of material used.
Find:

Sides of base in cm
Height in cm

Q.

If $x^3-2x^2{y}^2+5x+y-5=0$ and ${\left(y\right)}_{x=1}=1$, then

1. $y^{}\left(1\right)=\frac{2}{3}$

2. $y^{}\left(1\right)=\frac{4}{3}$

3. $y^{}\left(1\right)=-8\left(\frac{22}{7}\right)$

4. $y^{}\left(1\right)=-\frac{4}{3}$

Q.

Anshuman is preparing to make an ice sculpture. He has a block of ice of dimensions $3$ feet $\times$ $4$ feet $\times$$5$ feet that he wants to reduce in size by shaving off the same amount from the lengths, width, and height. He wants to reduce the volume of the ice block to $24$ cubic feet. Then the polynomial equation to model this situation is

1. $x^3-15x^2+47x+36=0$

2. $-x^3+47x^2-12x+38=0$

3. $-x^3+12x^2-47x+36=0$

4. $x^3+12x^2+47x-36=0$

Q. Find the derivative of $y=\frac{(t^2-1)}{(t^2+1)}$.

1. $\frac{4t^2}{(t+1{)}^2}$
2. $\frac{4t}{(t^2+1{)}^3}$
3. $\frac{4t}{(t+1{)}^2}$
4. $\frac{4t}{(t^2+1{)}^2}$

Q. Find the maximum and minimum value of function $f\left(x\right)=x^3-6x^2+9x+1$ in the interval $x=0\text{to}x=5$.

1. 20 and 2
2. 16 and 14
3. 18 and 2
4. 21 and 1