Page:Calculus Made Easy.pdf/53

Exercises II. (See p. 254 for Answers.)

Differentiate the following:

(1) ${\displaystyle y=ax^{3}+6}$.	(2) ${\displaystyle y=13x^{\tfrac {3}{2}}-c}$.
(3) ${\displaystyle y=12x^{\tfrac {1}{2}}+c^{\tfrac {1}{2}}}$.	(4) ${\displaystyle y=c^{\tfrac {1}{2}}x^{\tfrac {1}{2}}}$.
(5) ${\displaystyle u={\frac {az^{n}-1}{c}}}$.	(6) ${\displaystyle y=1.18t^{2}+22.4}$.

Make up some other examples for yourself, and try your hand at differentiating them.

(7) If ${\displaystyle l_{t}}$ and ${\displaystyle l_{0}}$ be the lengths of a rod of iron at the temperatures ${\displaystyle t^{\circ }}$ C. and ${\displaystyle 0^{\circ }}$ C. respectively, then ${\displaystyle l_{t}=l_{0}(1+0.000012t)}$. Find the change of length of the rod per degree Centigrade.

(8) It has been found that if ${\displaystyle c}$ be the candle power of an incandescent electric lamp, and ${\displaystyle V}$ be the voltage, ${\displaystyle c=aV^{b}}$, where ${\displaystyle a}$ and ${\displaystyle b}$ are constants.

Find the rate of change of the candle power with the voltage, and calculate the change of candle power per volt at ${\displaystyle 80}$, ${\displaystyle 100}$ and ${\displaystyle 120}$ volts in the case of a lamp for which ${\displaystyle a=0.5\times 10^{-10}}$ and ${\displaystyle b=6}$.

(9) The frequency ${\displaystyle n}$ of vibration of a string of diameter ${\displaystyle D}$, length ${\displaystyle L}$ and specific gravity ${\displaystyle \sigma }$, stretched with a force ${\displaystyle T}$, is given by

Find the rate of change of the frequency when ${\displaystyle D}$, ${\displaystyle L}$, ${\displaystyle \sigma }$ and ${\displaystyle T}$ are varied singly.