Differentiation from 1st Principles

Q. The area $A$ of a circle is related to its diameter by the equation $A=\frac{\pi }{4}{D}^2$. The change in area with respect to the diameter is $x\times \pi$, when the diameter is $10m$. Find the value of $x$.

Q. Find the derivative of functions $cosx$, $cosecx$, $secx$, $cotx$ using first principle​​​​​​

1. $-sinx,-cosecxcotx,secxtanx,-cose{c}^{2}x$
2. $sinx,-cosecxcotx,secxtanx,-cose{c}^{2}x$
3. $-sinx,-cosecxcotx,-secxtanx,cose{c}^{2}x$
4. $-sinx,cosecxcotx,-secxtanx,cose{c}^{2}x$

Q. If the displacement of a particle at any time $t$ is given by $s=ut+\frac{1}{2}a{t}^2$, where $u$ and $a$ are constants. Then the acceleration of particle at any time $t$ will be

1. $a$
2. $u+a$
3. $u$
4. None

Q. The derivative of a constant is some constant.

1. False
2. True

Q. If the displacement of particle at any time $t$ is given by $s=ut+\frac{1}{2}a{t}^2$ where $u$ is initial velocity (constant) and $a$ is acceleration (constant) then, velocity of particle at any time $t$ will be

1. $u+\frac{at}{2}$
2. $u+at$
3. $u+2at$
4. $\frac{u{t}^2}{2}+\frac{a{t}^3}{6}$

Q. The area $A$ of a circle is related to its diameter by the equation $A=\frac{\pi }{4}{D}^2$. The change in area with respect to the diameter is $x\times \pi$, when the diameter is $10m$. Find the value of $x$.

Q.

A curve is represented by y=sin x. If x is changed from $\frac{\pi }{3}to\frac{\pi }{3}+\frac{\pi }{100}$, find approximately the change in y.

1. $\frac{\pi }{100}$

2. $\frac{\pi }{200}$

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

4. None of these

Q. If the displacement of particle at any time $t$ is given by $s=ut+\frac{1}{2}a{t}^2$ where $u$ is initial velocity (constant) and $a$ is acceleration (constant) then, velocity of particle at any time $t$ will be

1. $u+\frac{at}{2}$
2. $u+at$
3. $u+2at$
4. $\frac{u{t}^2}{2}+\frac{a{t}^3}{6}$