Derivative of Standard Inverse Trigonometric Functions

Q.

If $y=t^{\frac{4}{3}}-3t^{\frac{-2}{3}}$, then $\frac{dy}{dt}=$

1. $\frac{2t^2+3}{3t^{{\displaystyle \frac{5}{3}}}}$

2. $\frac{2t^2+3}{t^{{\displaystyle \frac{5}{3}}}}$

3. $\frac{2(2t^2+3)}{t^{{\displaystyle \frac{5}{3}}}}$

4. $\frac{2(2t^2+3)}{3t^{{\displaystyle \frac{5}{3}}}}$

Q.

If $y=\sin \left(\frac{1+x^2}{1-x^2}\right)$, then $\frac{dy}{dx}=$

1. $\left(\frac{4x}{1-x^2}\right)\cos \left(\frac{1+x^2}{1-x^2}\right)$

2. $\left(\frac{x}{1-x^2}\right)\cos \left(\frac{1+x^2}{1-x^2}\right)$

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

4. $\frac{x}{{\left(1-x^2\right)}^2}\cos \left(\frac{1+x^2}{1-x^2}\right)$

Q.

The normal to the curve $y={\left(1+2x\right)}^x+{\sin }^{-1}\left({\sin }^3\left(x\right)\right)$ at $x=0$ is:

1. $y=0$

2. $x+y=1$

3. $x=0$

4. None of these

Q.

${\int }_{-1}^2{\left|x\right|}^{3}dx=$

1. $\frac{5}{4}$

2. $\frac{17}{4}$

3. $\frac{15}{4}$

4. $\frac{4}{5}$

Q. If for $x\in (0,\frac{1}{4})$, the derivative of ${\tan }^{-1}\left(\frac{6x\sqrt{x}}{1-9x^3}\right)$ is $\sqrt{x}\cdot g\left(x\right)$, then $g\left(x\right)$ equals:

1. $\frac{9}{1+9x^3}$
2. $\frac{3x\sqrt{x}}{1-9x^3}$
3. $\frac{3x}{1-9x^3}$
4. $\frac{3}{1+9x^3}$

Q. If $\underset{x\to a}{lim}\left[{\sin }^{-1}\frac{2x}{1+x^2}\right]$ doesn't exist, then the number of possible value(s) of $a$ is
(Here, $[.]$ denotes the greatest integer function)

Q. If $y=\sum _{r=1}^x{\tan }^{-1}\left(\frac{1}{1+r+r^2}\right),$ then $\frac{dy}{dx}$ at $x=2$ is equal to

1. $\frac{1}{5}$
2. $\frac{1}{10}$
3. $0$
4. $\frac{1}{2}$

Q. If for $x\in (0,\frac{1}{4})$, the derivative of ${\tan }^{-1}\left(\frac{6x\sqrt{x}}{1-9x^3}\right)$ is $\sqrt{x}\cdot g\left(x\right)$, then $g\left(x\right)$ equals:

1. $\frac{9}{1+9x^3}$
2. $\frac{3x\sqrt{x}}{1-9x^3}$
3. $\frac{3x}{1-9x^3}$
4. $\frac{3}{1+9x^3}$

Q. Match the following functions to their derivatives?
$\begin{array}{cc}Function& Derivatives\\ a)si{n}^{-1}x& 1)\frac{-1}{|x|\sqrt{x^2-1}}\\ b)co{s}^{-1}x& 2)\frac{-1}{1+x^2}\\ c)ta{n}^{-1}x& 3)\frac{1}{|x|\sqrt{x^2-1}}\\ d)se{c}^{-1}x& 4)\frac{1}{1+x^2}\\ e)co{t}^{-1}x& 5)\frac{-1}{\sqrt{1-x^2}}\\ f)cose{c}^{-1}x& 6)\frac{1}{\sqrt{1-x^2}}\end{array}$

1. a-6, b-5, c-4, d-3, e-2, f-1
2. a-5, b-6, c-4, d-3, e-2, f-1
3. a-6, b-5, c-3, d-4, e-2, f-1
4. a-6, b-5, c-3, d-2, e-1, f-2