y=a-xa-x+b-xx-ba-x-x-b, then dydx where ever it is defined is

y=a-x32-(x-b)32(a-x)12-(x-b)12 y=a-x+x-b-(a-x)(x-b) y=(a-b)-a-xx-b dydx=x-b2a-x-a-x2x-b =x-b-a+x2a-xx-b =2x-(a+b)2a-xx-b

y=a-xa-x+b-xx-ba-x-x-b, then dydx where ever it is defined is

y=a-x32-(x-b)32(a-x)12-(x-b)12 y=a-x+x-b-(a-x)(x-b) y=(a-b)-a-xx-b dydx=x-b2a-x-a-x2x-b =x-b-a+x2a-xx-b =2x-(a+b)2a-xx-b

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$y = \frac{(a - x) \sqrt{a - x} + (b - x) \sqrt{x - b}}{\sqrt{a - x} - \sqrt{x - b}},$ then $\frac{dy}{dx}$ where ever it is defined is

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

$\frac{2 x + a + b}{2 \sqrt{(a - x) (x - b)}}$

$\frac{2 x - a - b}{2 \sqrt{a - x} \sqrt{a - b}}$

$\frac{a + b}{2 \sqrt{(a - x) (x - b)}}$

answer is B.

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Detailed Solution

$y = \frac{{(a - x)}^{\frac{3}{2}} - {(x - b)}^{\frac{3}{2}}}{{(a - x)}^{\frac{1}{2}} - {(x - b)}^{\frac{1}{2}}} y = (a - x + x - b) - \sqrt{(a - x) (x - b)} y = (a - b) - \sqrt{a - x} \sqrt{x - b} \frac{dy}{dx} = \frac{\sqrt{x - b}}{2 \sqrt{a - x}} - \frac{\sqrt{a - x}}{2 \sqrt{x - b}} = \frac{x - b - a + x}{2 \sqrt{(a - x) (x - b)}} = \frac{2 x - (a + b)}{2 \sqrt{(a - x) (x - b)}}$