Quotient Rule of Differentiation

Q.

$\cos ec\left[2co{t}^{–1}\left(5\right)+{\cos }^{–1}\left(\frac{4}{5}\right)\right]$ is equal to

1. $\frac{75}{56}$

2. $\frac{65}{56}$

3. $\frac{56}{33}$

4. $\frac{65}{33}$

Q.

The value of $\underset{x\to 0}{\lim }\frac{\left[1-\cos \left(2x\right)\right]}{x^2}$ is

1. dose not exist.

2. infinite

3. $0$

4. $2$

Q.

If $co{t}^{-1}\left(ɑ\right)=co{t}^{-1}2+co{t}^{-1}8+co{t}^{-1}18+co{t}^{-1}32+\dots \dots ..$ upto $100$ terms, then $ɑ$ is:

1. $1.03$

2. $1.00$

3. $1.01$

4. $1.02$

Q.

$\int \left\{\frac{\left(secx\right)}{\left(secx+\tan x\right)}\right\}dx=$

1. $\tan x-secx+c$

2. $\log \left(1+secx\right)+c$

3. $secx+\tan x+c$

4. $\log \sin x+\log \cos x+c$

Q. If $y=\frac{sinx+cosx}{sinx-cosx}$, then $\frac{dy}{dx}$ at $x=0$ is

1. -2
2. 0
3. 1/2
4. does not exist

Q.

Evaluate $\tan \left[\left(\frac{1}{2}\right){\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]=$

(a) $\frac{3-\sqrt{5}}{2}$

(b) $\frac{3+\sqrt{5}}{2}$

(c) $\frac{2}{3-\sqrt{5}}$

(d) $\frac{2}{3+\sqrt{5}}$

1. $aandd$

2. $donly$

3. $bonly$

4. $Noneofthese$

Q.

Differentiate the following functions with respect to x :

$\frac{x}{1+tanx}$

Q.

Evaluate $\tan \left[{\sec }^{-1}\left(\sqrt{1+x^2}\right)\right]$

1. $\frac{1}{x}$

2. $x$

3. $\frac{1}{\sqrt{1+x^2}}$

4. $\frac{x}{\sqrt{1+x^2}}$

Q.

If $x,y,z$ are in A.P and ${\tan }^{-1}x,{\tan }^{-1}y\mathrm{and}{\tan }^{-1}z$ are also in A.P., then

1. $x=y=z$

2. $x=y=-z$

3. $x=1,y=2,z=3$

4. $x=2,y=4,z=6$

Q.

$\int (1-\cos x)\cos e{c}^{2}xdx=$

1. $\tan \left(\frac{x}{2}\right)+c$

2. $–cot\left(\frac{x}{2}\right)+c$

3. $2\tan \left(\frac{x}{2}\right)+c$

4. $–2cot\left(\frac{x}{2}\right)+c$

Q.

$\frac{d}{dx}\left[\left(\frac{{\tan }^2\left(2x\right)-{\tan }^2\left(x\right)}{1-{\tan }^2\left(2x\right){\tan }^2\left(x\right)}\right)\cot \left(3x\right)\right]=$

1. ${\sec }^2\left(x\right)$

2. $\sec \left(x\right)\tan \left(x\right)$

3. $\tan \left(2x\right)\tan \left(x\right)$

4. none of these

Q.

The value of $\frac{d}{dx}\left[{\tan }^{-1}\frac{\sqrt{x}(3-x)}{(1-3x)}\right]$is

1. $\frac{1}{2(1+x)\sqrt{x}}$

2. $\frac{3}{(1+x)\sqrt{x}}$

3. $\frac{2}{(1+x)\sqrt{x}}$

4. $\frac{3}{2(1+x)\sqrt{x}}$

Q.

Let $n$ be a positive integer such that $\sin \left(\frac{\pi }{2n}\right)+\cos \left(\frac{\pi }{2n}\right)=\frac{\sqrt{n}}{2}$. Then

1. $6\le n\le 8$

2. $4<n\le 8$

3. $4\le n\le 8$

4. $4<n<8$

Q.

The value of $\cos \left(52°\right)+\cos \left(68°\right)+\cos \left(172°\right)$ is

1. $0$

2. $1$

3. $2$

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

Q. If $y=\frac{1+\frac{1}{x^2}}{1-\frac{1}{x^2}},$ then $\frac{dy}{dx}=$

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

Q.

$\frac{d\left({\tan }^{-1}\left({\displaystyle \frac{4\sqrt{x}}{1-4x}}\right)\right)}{dx}=$

1. $\frac{1}{\sqrt{x}\left(1+4x\right)}$

2. $\frac{2}{\sqrt{x}\left(1+4x\right)}$

3. $\frac{1}{\sqrt{x}\left(1+4x\right)}$$x$

4. none of these

Q.

The maximum value of $5\cos \left(\mathrm{\theta }\right)+3\cos \left(\mathrm{\theta }+\frac{\mathrm{\pi }}{3}\right)+3$ is

1. $5$

2. $11$

3. $10$

4. $-11$

Q.

Differentiate the following functions with respect to x :

$\frac{1+logx}{1-logx}$

Q.

If ${\tan }^2\left(\theta \right)=2\tan \left(\varphi \right)+1$, then $\cos \left(2\theta \right)+{\sin }^2\left(\varphi \right)$ equals

1. $-1$

2. $0$

3. $1$

4. none of these

Q.

If $\tan \left(\frac{\theta }{2}\right)=t$, then $\frac{1-t^2}{1+t^2}$ is equal to

1. $\cos \left(\theta \right)$

2. $\sin \left(\theta \right)$

3. $\sec \left(\theta \right)$

4. $\cos \left(2\theta \right)$

Q.

The value of $\underset{x\to \frac{\pi }{2}}{\lim }{\left(\sin \left(x\right)\right)}^{\tan \left(x\right)}$ is

1. $1$

2. $0$

3. $e$

4. None of these.

Q.

Differentiate the following functions with respect to x :

$\frac{x}{si{n}^{n}x}$

Q.

The value of $\frac{cotx-\tan x}{cot2x}$ is

1. $1$

2. $2$

3. $-1$

4. $4$

Q.

Differentiate: $se{c}^{-1}\left(x\right)$

Q. Find the derivative of $(ax+b)(cx+d{)}^2$ where $a,b,c,d$ are fixed non-zero constants.

Q.

The derivative of $\sqrt{\left(\sqrt{x}+1\right)}$ is

1. $\frac{1}{\sqrt{x}\sqrt{\left(\sqrt{\mathrm{x}}+1\right)}}$

2. $\frac{1}{\sqrt{x}\left(\sqrt{\mathrm{x}}+1\right)}$

3. $\frac{4}{\sqrt{x}\sqrt{\left(\sqrt{\mathrm{x}}+1\right)}}$

4. $\frac{1}{4\sqrt{\left(\mathrm{x}\sqrt{\mathrm{x}}+1\right)}}$

Q.

The equation $2{\cos }^2\frac{x}{2}{\sin }^{2}x=x^2+x^{-2};0<x\le \frac{\mathrm{\pi }}{2}$ has

1. no real solution

2. one real solution

3. more than one solution

4. none of these

Q.

Differentiate the following functions with respect to x :

$\frac{\sqrt{a}+\sqrt{x}}{\sqrt{a}-\sqrt{x}}$

Q.

Differentiate the following functions with respect to x :

$\frac{x^5-cosx}{sinx}$

Q.

Differentiate the following functions with respect to $x$ :

$\frac{{10}^x}{\sin x}$