Expand the numerator by the binomial theorem (see p. 141).

hence ${\displaystyle {\dfrac {dP}{dt}}={\dfrac {1}{537,824\times 10^{5}}}}$,

when ${\displaystyle t=100}$ this becomes ${\displaystyle 0.036}$ atmosphere per degree Centigrade change of temperature.

Exercises III. (See the Answers on p. 255.)

(1) Differentiate

(a) ${\displaystyle u=1+x+{\dfrac {x^{2}}{1\times 2}}+{\dfrac {x^{3}}{1\times 2\times 3}}+\dotsb }$

(b) ${\displaystyle y=ax^{2}+bx+c}$

(c) ${\displaystyle y=(x+a)^{2}}$

(d) ${\displaystyle y=(x+a)^{3}}$

(2) If ${\displaystyle w=at-{\frac {1}{2}}bt^{2}}$, find ${\displaystyle {\dfrac {dw}{dt}}}$.

(3) Find the differential coefficient of

(4) Differentiate

(5) If ${\displaystyle x=(y+3)\times (y+5)}$, find ${\displaystyle {\dfrac {dx}{dy}}}$.

(6) Differentiate ${\displaystyle y=1.3709x\times (112.6+45.202x^{2})}$