Question

Find dy/dx
y = 1 + x/(x-a) + bx/(x-a)(x-b) + cx²/(x-a)(x-b)(x-c) .

Answer 1

$y = \huge{(} \small{1 + \dfrac{ a}{x - a} } \huge{)} \small{ + \huge{(} \small{ \dfrac{ bx}{(x-a)(x-b)} + \dfrac{ cx^{2}}{( x - a)(x-b)(x-c)} }} \huge{)}$




$y = \dfrac{x - a + a}{x - a} + \dfrac{bx(x - c) + cx {}^{2} }{(x - a)(x - b)(x - c)} \\ \\ \\ \\ y = \frac{x}{x - a} + \frac{b {x}^{2} - cbx + cx {}^{2} }{(x - a)(x - b)(x - c)} \\ \\ \\ \\ = > y = \frac{x(x - b)(x - c) + bx {}^{2} - cbx + cx {}^{2} }{ (x - a)(x - b)(x - c)}$



$= >y = \dfrac{ {x}^{3} - {x}^{2} c - bx {}^{2} + bcx + bx {}^{2} - bcx + cx {}^{2} }{(x - a)(x - b)(x - c)} \\ \\ \\ \\ = > y = \frac{ {x}^{3} }{(x - a)(x - b)(x - c)}$





Taking log on both sides,


$= > log \: y = log \: \dfrac{ {x}^{3} }{(x - a)(x - b)(x - c)}$


From the properties of logarithms ,



$= > log \: y = log {x}^{3} - log(x - a) - log(x - b) - log(x - c)$



From the properties of logarithms ,



$= > log \: y = 3log {x}^{} - log(x - a) - log(x - b) - log(x - c)$



Then,

$= > \dfrac{ y_{1}}{y} = \dfrac{3}{x} + \dfrac{1}{x - a} + \dfrac{1}{x - b} + \dfrac{1}{x - c} \\ \\ \\ \\ = > \frac{dy}{dx} = y \times ( \dfrac{ 1 }{ x } - \frac{1}{x - a} + \frac{1}{x} - \frac{1}{x - b} + \frac{1}{x} - \frac{1}{x - c} )$
$\\ \\ \\ = > \frac{dy}{dx} = y \times ( \frac{ x - a - x}{x(x - a)} + \frac{x - b - x}{x(x - b) } + \frac{x - c - x }{x(x - c)} \\ \\ \\ \\ = > \frac{dy}{dx} = \frac{y}{x} \times ( \frac{ - a}{x - a} + \frac{ - b}{x - b} + \frac{ - c}{x - c} ) \\ \\ \\ \\ = > \frac{dy}{dx} = \frac{y}{x} \times ( \frac{a}{a - x} + \frac{b}{b - x} + \frac{c}{c - x} )$






Please check it.

Answer 2

Step-by-step explanation:

Then,

$\begin{lgathered} \sf \: = > \dfrac{ y_{1}}{y} = \dfrac{3}{x} + \dfrac{1}{x - a} + \dfrac{1}{x - b} + \dfrac{1}{x - c} \\ \\ \\ \\ \sf \: = > \frac{dy}{dx} = y \times ( \dfrac{ 1 }{ x } - \frac{1}{x - a} + \frac{1}{x} - \frac{1}{x - b} + \frac{1}{x} - \frac{1}{x - c} ) \\\end{lgathered} </p><p>$

$\begin{lgathered}\\ \\ \\ \sf \: = > \frac{dy}{dx} = y \times ( \frac{ x - a - x}{x(x - a)} + \frac{x - b - x}{x(x - b) } + \frac{x - c - x }{x(x - c)} \\ \\ \\ \\ \sf \: = > \frac{dy}{dx} = \frac{y}{x} \times ( \frac{ - a}{x - a} + \frac{ - b}{x - b} + \frac{ - c}{x - c} ) \\ \\ \\ \\ = > \sf \red{\frac{dy}{dx} = \frac{y}{x} \times ( \frac{a}{a - x} + \frac{b}{b - x} + \frac{c}{c - x} )}\end{lgathered}$