Page:Calculus Made Easy.pdf/66

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

$P={\frac {1}{140^{5}}}(40^{5}+5\times 40^{4}t+10\times 40^{3}t^{2}+10\times 40^{2}t^{3}+5\times 40t^{4}+t^{5})$;

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

$(5\times 40^{4}+20\times 40^{3}t+30\times 40^{2}t^{2}+20\times 40t^{3}+5t^{4})$,

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

 

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

(1) Differentiate

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

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

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

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

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

(3) Find the differential coefficient of

$y=(x+{\sqrt {-1}})\times (x-{\sqrt {-1}})$.

(4) Differentiate

$y=(197x-34x^{2})\times (7+22x-83x^{3}).$.

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

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