Derivative of Some Standard Functions

Q.

If $p=\cos {55}^{\circ }$ , $q=\cos {65}^{\circ }$ and $r=\cos {175}^{\circ }$ then the value of $\frac{1}{p}+\frac{1}{q}+\frac{r}{pq}$ is

1. $0$

2. $-1$

3. $1$

4. None of these

Q.

Describe Inverse Trigonometric Functions

Q.

$f\left(x\right)$ has third continuous derivative and $\underset{x\to 0}{\lim }{\left[1+x+\frac{f\left(x\right)}{x}\right]}^{\frac{1}{x}}=e^3$, then $f\left(x\right)$ is

1. $x^2$

2. $3x^2$

3. $2x^2$

4. None of these

Q. Suppose $f\left(x\right)=\frac{x-1}{2x^2-7x+5}$ for $x\ne 1$ and $f\left(1\right)=-\frac{1}{3}$, then

1. $f$ is continuous but not differentiable at $x=1$
2. $f$ is differentiable $x=1$ and $f^{\prime }\left(1\right)=-\frac{1}{3}$
3. $f$ is differentiable $x=1$ and $f^{\prime }\left(1\right)=-\frac{2}{9}$
4. $f$ is discontinuous at $x=\frac{5}{2}$

Q. ${\Delta }_1=\left|\begin{array}{ccc}x& b& b\\ a& x& b\\ a& a& x\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{cc}x& b\\ a& x\end{array}\right|$ are the given determinants, then

1. ${\Delta }_1=3({\Delta }_2{)}^2$
2. $\frac{d}{dx}\left({\Delta }_1\right)=3{\Delta }_2$
3. $\frac{d}{dx}\left({\Delta }_1\right)=3({\Delta }_2{)}^2$
4. ${\Delta }_1=3({\Delta }_2{)}^{3/2}$

Q. If $f(x-y),f\left(x\right)f\left(y\right)$ and $f(x+y)$ are in A.P. for all $x,y\in R$ and $f\left(0\right)\ne 0$, then

1. $f\left(4\right)=f(-4)$
2. $f\left(2\right)+f(-2)=0$
3. $f^{\prime }\left(4\right)+f^{\prime }(-4)=0$
4. $f^{\prime }\left(2\right)=f^{\prime }(-2)$

Q.

How do you find the derivative of $y=\frac{1}{\sqrt{x}}$ ?

Q.

$\frac{d}{dx}\left(\frac{1}{x\sqrt{x}}\right)=$

1. $\frac{-3}{2}{x}^{-3/2}$

2. $\frac{-3}{2}{x}^{-5/2}$

3. $x^{-5/2}$

4. None of the above

Q. If $y\left(\alpha \right)=\sqrt{2\left(\frac{\tan \alpha +\cot \alpha }{1+{\tan }^2\alpha }\right)+\frac{1}{{\sin }^2\alpha }}$ where $\alpha \in (\frac{3\pi }{4},\pi )$, then $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\pi }{6}$ is

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

Q.

$\frac{d}{dx}\left(\frac{1}{x\sqrt{x}}\right)=$

1. $\frac{-3}{2}{x}^{-3/2}$

2. $\frac{-3}{2}{x}^{-5/2}$

3. $x^{-5/2}$

4. None of the above

Q. Match the following

$\begin{array}{cc}Functions& Derivatives\\ \left(a\right)\sin x& 1)-\sin x\\ \left(b\right)\cos x& 2){\sec }^{2}x\\ \left(c\right)\tan x& 3)\cos x\\ \left(d\right)\sec x& 4)-cose{c}^{2}x\\ \left(e\right)\cot x& 5)\sec x\tan x\\ \left(f\right)cosecx& 6)-cosecxcotx\end{array}$

1. a – 1, b – 3, c – 2, d – 4, e – 5, f – 6
2. a – 3, b – 1, c – 2, d – 5, e – 4, f – 6
3. a – 2, b – 1, c – 3, d – 5, e – 4, f – 6
4. None of the above