Differentiation from 1st Principles

Q. The derivative of a constant is some constant.

1. False
2. True

Q. Find the derivative of $x^2$ using first principle.

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

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