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Class-12-science»Maths

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We have, y = \sqrt{(a - x) (x - b)} - (a - b) \tan^{- 1} \sqrt{\frac{a - x}{x - b}} \Rightarrow \frac{dy}{dx} = \frac{a + b - 2 x}{2 \sqrt{(a - x) (x - b)}} - (a - b) \times \frac{1}{1 + (\frac{a - x}{x - b})} \times \frac{\sqrt{x - b}}{2 \sqrt{a - x}} \times \frac{(x - b) (- 1) - (a - x) (1)}{{(x - b)}^{2}} \Rightarrow \frac{dy}{dx} = \frac{a + b - 2 x}{2 \sqrt{(a - x) (x - b)}} - (a - b) \times \frac{(x - b)}{(a - b)} \times \frac{\sqrt{x - b}}{2 \sqrt{a - x}} \times \frac{(b - a)}{{(x - b)}^{2}} \Rightarrow \frac{dy}{dx} = \frac{a + b - 2 x}{2 \sqrt{(a - x) (x - b)}} - \frac{(b - a)}{2 \sqrt{(a - x) (x - b)}} \Rightarrow \frac{dy}{dx} = \frac{a + b - 2 x - b + a}{2 \sqrt{(a - x) (x - b)}} = \frac{2 (a - x)}{2 \sqrt{(a - x) (x - b)}} = \sqrt{\frac{(a - x)}{(x - b)}} \Rightarrow \frac{dy}{dx} = \sqrt{\frac{(a - x)}{(x - b)}}

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