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130

Calculus Made Easy

Application to differentiation. Let it be required to differentiate $y={\dfrac {5-4x}{6x^{2}+7x-3}}$ ; we have

${\begin{aligned}{\frac {dy}{dx}}&=-{\frac {(6x^{2}+7x-3)\times 4+(5-4x)(12x+7)}{(6x^{2}+7x-3)^{2}}}\\&={\frac {24x^{2}-60x-23}{(6x^{2}+7x-3)^{2}}}.\end{aligned}}$

If we split the given expression into

${\frac {1}{3x-1}}-{\frac {2}{2x+3}}$ ,

we get, however,

${\frac {dy}{dx}}=-{\frac {3}{(3x-1)^{2}}}+{\frac {4}{(2x+3)^{2}}}$ ,

which is really the same result as above split into partial fractions. But the splitting, if done after differentiating, is more complicated, as will easily be seen. When we shall deal with the integration of such expressions, we shall find the splitting into partial fractions a precious auxiliary (see p. 230).

Exercises XI. (See p. 259 for Answers.)

Split into fractions:

(1) ${\dfrac {3x+5}{(x-3)(x+4)}}$ .

(2) ${\dfrac {3x-4}{(x-1)(x-2)}}$ .

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

(4) ${\dfrac {x+1}{x^{2}-7x+12}}$ .

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

(6) ${\dfrac {x^{2}-13x+26}{(x-2)(x-3)(x-4)}}$ .

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