Page:Calculus Made Easy.pdf/151

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OTHER USEFUL DODGES

131

(7) ${\dfrac {x^{2}-3x+1}{(x-1)(x+2)(x-3)}}$ .

(8) ${\dfrac {5x^{2}+7x+1}{(2x+1)(3x-2)(3x+1)}}$ .

(9) ${\dfrac {x^{2}}{x^{3}-1}}$ .

(10) ${\dfrac {x^{4}+1}{x^{3}+1}}$ .

(11) ${\dfrac {5x^{2}+6x+4}{(x+1)(x^{2}+x+1)}}$ .

(12) ${\dfrac {x}{(x-1)(x-2)^{2}}}$ .

(13) ${\dfrac {x}{(x^{2}-1)(x+1)}}$ .

(14) ${\dfrac {x+3}{(x+2)^{2}(x-1)}}$ .

(15) ${\dfrac {3x^{2}+2x+1}{(x+2)(x^{2}+x+1)^{2}}}$ .

(16) ${\dfrac {5x^{2}+8x-12}{(x+4)^{3}}}$ .

(17) ${\dfrac {7x^{2}+9x-1}{(3x-2)^{4}}}$ .

(18) ${\dfrac {x^{2}}{(x^{3}-8)(x-2)}}$ .

Differential of an Inverse functions.

Consider the function (see p. 14) $y=3x$ ; it can be expressed in the form $x={\dfrac {y}{3}}$ ; this latter form is called the inverse function to the one originally given.

If $y=3x$ , ${\dfrac {dy}{dx}}=3$ ; if $x={\dfrac {y}{3}}$ , ${\dfrac {dx}{dy}}={\dfrac {1}{3}}$ , and we see that

${\frac {dy}{dx}}={\frac {1}{\ {\dfrac {dx}{dy}}\ }}\quad {\text{or}}\quad {\frac {dy}{dx}}\times {\frac {dx}{dy}}=1$ .

Consider $y=4x^{2}$ , ${\dfrac {dy}{dx}}=8x$ ; the inverse function is

$x={\frac {y^{\frac {1}{2}}}{2}},\quad {\text{and}}\quad {\frac {dx}{dy}}={\frac {1}{4{\sqrt {y}}}}={\frac {1}{4\times 2x}}={\frac {1}{8x}}$ .