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46

Calculus Made Easy

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.

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