Derivative of Standard Functions

Q.

If $\mathrm{y}={\mathrm{x}}^{{\mathrm{x}}^{\mathrm{x}....\infty }},$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

1. ${\mathrm{yx}}^{\mathrm{y}-1}$

2. $\frac{{\mathrm{y}}^2}{\mathrm{x}(1-\mathrm{ylog}\mathrm{x})}$

3. $\frac{\mathrm{y}}{\mathrm{x}\left(1+\mathrm{ylogx}\right)}$

4. None of these

Q.

What are the formulae of $\left(1\right)1+\cos 2x\left(2\right)1-\cos 2x$.

Q. Find the value of d/ dx(√x+1/√x)^2

Q.

If $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\left(1-\mathrm{x}\right)}$, then the derivative of the composite function $\mathrm{f}\left[\mathrm{f}\left\{\mathrm{f}\left(\mathrm{x}\right)\right\}\right]$ is

1. 0

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

3. 1

4. 2

Q.

If $A=\left[\begin{array}{cc}2& 2\\ 9& 4\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ then $10A^{-1}$ is equal to:

1. $A+6I$

2. $A-6I$

3. $A+4I$

4. $A-4I$

Q.

The sum of ${162}^{\mathrm{th}}$ power of the roots of the equation $x^3-2x^2+2x-1=0$ is

Q.

$\frac{d}{dx}\left({\tan }^{-1}\left(\sqrt{\frac{1+\cos \left({\displaystyle \frac{x}{2}}\right)}{1-\cos \left({\displaystyle \frac{x}{2}}\right)}}\right)\right)$ is equal to

1. $-\frac{1}{4}$

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

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

4. $-\frac{1}{2}$

Q.

$\int \left[\frac{f\left(x\right)}{\sqrt{f\left(x\right)}}\right]dx=\_\_\_\_\_\_\_+c$; $f\left(x\right)\ne 0$

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

2. $2\sqrt{f\left(x\right)}$

3. $\left(\frac{1}{2}\right)f\left(x\right)$

4. $2f\left(x\right)$

Q.

If $y={\sin }^{-1}\sqrt{x}$, then $\frac{dy}{dx}=$

1. $\frac{2}{\sqrt{x}\sqrt{1-x}}$

2. $\frac{-2}{\sqrt{x}\sqrt{1-x}}$

3. $\frac{1}{2\sqrt{x}\sqrt{1-x}}$

4. $\frac{1}{\sqrt{1-x}}$

Q.

The range of the function $f\left(x\right)=tan\sqrt{\left(\frac{{\pi }^2}{9}\right)-x^2}$ is

1. $[0,3]$

2. $[0,\surd 3]$

3. $(-\infty ,\infty )$

4. None of these

Q.

What is the formula of $\cos \left(4x\right)$?

Q.

The domain of the function $f\left(x\right)=\frac{1}{\sqrt{\left|x\right|-x}}$ is

1. $\left(-\infty ,\infty \right)$

2. $\left(0,\infty \right)$

3. $\left(-\infty ,0\right)$

4. $\left(-\infty ,\infty \right)-\left\{0\right\}$

Q.

Let $\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}-\left[\mathrm{x}\right]/1+\mathrm{x}-\left[\mathrm{x}\right]$ where $\left[\mathrm{x}\right]$ denotes the greatest integer less than or equal to $\mathrm{x}$ then the range of $\mathrm{f}$ is

1. $[0,1]$

2. $[0,1/2)$

3. $[0,1)$

4. $[0,1/2]$

Q.

$\frac{d}{dx}{\cos }^{-1}\sqrt{\cos x}$

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

2. $\sqrt{(secx+1)}$

3. $-\left(\frac{1}{2}\right)\sqrt{(secx+1)}$

4. $-\sqrt{(secx+1)}$

Q.

If $\sin x+\cos ecx=2$ then ${\sin }^{n}x+\cos e{c}^{n}x$ is equal to

1. $2$

2. $2^n$

3. $2^{n–1}$

4. $2^{n–2}$

Q. The points at which the function $f\left(x\right)=\frac{x+1}{x^2+x-12}$ is discontinuous, are

1. –3, 4
2. 3, –4
3. –1, –3, 4
4. –1, 3, 4

Q.

For what value of $x$ the matrix $A$ is singular

$\left[\begin{array}{cc}1+x& 7\\ 3-x& 8\end{array}\right]$

Q.

For the equation ${\cos }^{-1}x+{\cos }^{-1}2x+\pi =0$, the number of real solution is

1. $1$

2. $2$

3. $0$

4. infinity

Q.

Find the derivative of $\cos \sqrt{x}$ with respect to $x$ using the first principle.

Q.

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

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

2. $0$

3. $1$

4. $-\frac{1}{2}$

Q.

If $\cos 2x=(\sqrt{2}+1)(\cos x-(\frac{1}{\sqrt{2}}\left)\right)$ then $\cos x\ne \frac{1}{\sqrt{2}},x\in$

1. $\left\{2n\pi \pm \frac{\pi }{3}:n\in Z\right\}$

2. $\left\{2n\pi \pm \frac{\pi }{6}:n\in Z\right\}$

3. $\left\{2n\pi \pm \frac{\pi }{2}:n\in Z\right\}$

4. $\left\{2n\pi \pm \frac{\pi }{4}:n\in Z\right\}$

Q.

If function $\mathrm{f}\left(\mathrm{x}\right)=\frac{1}{2}-\tan \left(\frac{\mathrm{\pi x}}{2}\right);(-1<\mathrm{x}<1)$ and $\mathrm{g}\left(\mathrm{x}\right)=\sqrt{3+4\mathrm{x}-4{\mathrm{x}}^2}$, then domain of $(f+g)$ is given by

1. $\left(\frac{1}{2},1\right)$

2. $\left(\frac{1}{2},-1\right)$

3. $\left(\frac{-1}{2},1\right)$

4. $\left(\frac{-1}{2},-1\right)$

Q. Which of the following are true for the function $f\left(x\right)={\log }_e(3x^2-4x+5)$ (where $D_f,R_f$ represents domain and range of function $f\left(x\right)$ reapectively)

1. $R_f=[{\log }_e\frac{11}{3},\infty )$
2. $D_f=R$
3. $D_f=[0,\infty )$
4. $R_f=R$

Q. Let $x\left(t\right)=2\sqrt{2}\cos t\sqrt{\sin 2t}$ and $y\left(t\right)=2\sqrt{2}\sin t\sqrt{\sin 2t},t\in (0,\frac{\pi }{2}).$ Then $\frac{1+{\left(\frac{dy}{dx}\right)}^2}{\frac{d^{2}y}{d{x}^2}}$ at $t=\frac{\pi }{4}$ is equal to

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

Q.

Find the domain of the function$f\left(x\right)=\sqrt{[\frac{2}{(x^2-x+1)}-\frac{1}{(x+1)}-\frac{2x-1}{(x^3+1)}]}$

1. $(-\infty ,2]-\{-1\}$

2. $(-\infty ,2)$

3. $]-1,2]$

4. None of these

Q.

Prove that the greatest integer function defined by f(x) = [x], 0 < x < 3 is not differentiable at x = 1 and x = 2.

Q. Let$f\left(x\right)=min\{1,1+x\sin x\},0\le x\le 2\pi$. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair $(m,n)$ is equal to

1. $(1,0)$
2. $(1,1)$
3. $(2,1)$
4. $(2,0)$

Q. In a triangle $ABC,$ If two sides are $a=6,b=3$ and $\cos (A-B)=\frac{4}{5}$, then the area of the triangle is:

1. $9$ Sq. units
2. $3$ Sq. units
3. $18$ Sq. units
4. $\frac{9}{2}$ Sq. units

Q. Let $a=min\{x^2+2x+3:x\in R\}$ and $b=\underset{\theta \to 0}{lim}\frac{1-\cos \theta }{{\theta }^2}.$ Then $\sum _{r=0}^n{a}^r{b}^{n-r}$ is

1. $\frac{1}{2}(2^n-1)$
2. $\frac{2^{n+1}-1}{3\cdot 2^n}$
3. $\frac{2^{n+1}+1}{3\cdot 2^n}$
4. $\frac{4^{n+1}-1}{3\cdot 2^n}$

Q.

If $|x|=5$ then$x$ is equal to ____.