If y=ax2x−ax−bx−c+bxx−bx−c+cx−c+1 then y'y=

y=ax2x−ax−bx−c+bxx−bx−c+c+x−cx−c=ax2x−ax−bx−c+bxx−bx−c+xx−c=ax2x−ax−bx−c+xx−cbx−b+1=ax2x−ax−bx−c+xx−cb+x−bx−b=ax2x−ax−bx−c+x2x−bx−c=ax2+x2x−ax−ax−bx−c=x3x−ax−bx−c∴log y=3logx−logx−a−logx−b−logx−cDifferentiating w.r.t. x,⇒1ydydx=3x−1x−a−1x−b−1x−c⇒y'y=3x−1x−a+1x−b+1x−c=1x−1x−a+1x−1x−b+1x−1x−c=−1xax−a+bx−b+cx−c=1xaa−x+bb−x+cc−x

If y=ax2x−ax−bx−c+bxx−bx−c+cx−c+1 then y'y=

y=ax2x−ax−bx−c+bxx−bx−c+c+x−cx−c=ax2x−ax−bx−c+bxx−bx−c+xx−c=ax2x−ax−bx−c+xx−cbx−b+1=ax2x−ax−bx−c+xx−cb+x−bx−b=ax2x−ax−bx−c+x2x−bx−c=ax2+x2x−ax−ax−bx−c=x3x−ax−bx−c∴log y=3logx−logx−a−logx−b−logx−cDifferentiating w.r.t. x,⇒1ydydx=3x−1x−a−1x−b−1x−c⇒y'y=3x−1x−a+1x−b+1x−c=1x−1x−a+1x−1x−b+1x−1x−c=−1xax−a+bx−b+cx−c=1xaa−x+bb−x+cc−x

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If $y = \frac{a x^{2}}{(x - a) (x - b) (x - c)} + \frac{b x}{(x - b) (x - c)} + \frac{c}{(x - c)} + 1$ then $\frac{y'}{y} =$

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$\frac{a}{x - a} + \frac{b}{x - b} + \frac{c}{x - c}$

$\frac{1}{x} (\frac{a}{a - x} + \frac{b}{b - x} + \frac{c}{c - x})$

$\frac{1}{x} (\frac{a}{x - a} + \frac{b}{x - b} + \frac{c}{x - c})$

$\frac{a}{a - x} + \frac{b}{b - x} + \frac{c}{c - x}$

answer is A.

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Detailed Solution

$\begin{matrix} y = \frac{a x^{2}}{(x - a) (x - b) (x - c)} + \frac{b x}{(x - b) (x - c)} + \frac{c + x - c}{x - c} \\ = \frac{a x^{2}}{(x - a) (x - b) (x - c)} + \frac{b x}{(x - b) (x - c)} + \frac{x}{x - c} \\ = \frac{a x^{2}}{(x - a) (x - b) (x - c)} + \frac{x}{x - c} [\frac{b}{x - b} + 1] \\ = \frac{a x^{2}}{(x - a) (x - b) (x - c)} + \frac{x}{x - c} [\frac{b + x - b}{x - b}] \\ = \frac{a x^{2}}{(x - a) (x - b) (x - c)} + \frac{x^{2}}{(x - b) (x - c)} \\ = \frac{a x^{2} + x^{2} (x - a)}{(x - a) (x - b) (x - c)} \\ = \frac{x^{3}}{(x - a) (x - b) (x - c)} \\ ∴ \log y = 3 \log x - \log (x - a) - \log (x - b) - \log (x - c) \\ D i f f e r e n t i a t i n g w . r . t . x, \\ \Rightarrow \frac{1}{y} \frac{d y}{d x} = \frac{3}{x} - \frac{1}{x - a} - \frac{1}{x - b} - \frac{1}{x - c} \\ \Rightarrow \frac{y'}{y} = \frac{3}{x} - (\frac{1}{x - a} + \frac{1}{x - b} + \frac{1}{x - c}) \\ = (\frac{1}{x} - \frac{1}{x - a}) + (\frac{1}{x} - \frac{1}{x - b}) + (\frac{1}{x} - \frac{1}{x - c}) \\ = - \frac{1}{x} (\frac{a}{x - a} + \frac{b}{x - b} + \frac{c}{x - c}) \\ = \frac{1}{x} (\frac{a}{a - x} + \frac{b}{b - x} + \frac{c}{c - x}) \end{matrix}$