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If √ Y + X + √ Y − X = C , Show that D Y D X = Y X − √ Y 2 X 2 − 1 ? - CBSE (Arts) Class 12 - Mathematics

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Question

If \[\sqrt{y + x} + \sqrt{y - x} = c, \text {show that } \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\] ?

Solution

\[\text{ Here,} \] \[ \sqrt{y + x} + \sqrt{y - x} = c\]

Differentiating with respect to x,

\[\Rightarrow \frac{d}{dx}\left( \sqrt{y + x} \right) + \frac{d}{dx}\sqrt{y - x} = \frac{d}{dx}\left( c \right)\]

\[ \Rightarrow \frac{1}{2\sqrt{y + x}}\frac{d}{dx}\left( y + x \right) + \frac{1}{2\sqrt{y - x}}\frac{d}{dx}\left( y - x \right) = 0 \]

\[ \Rightarrow \frac{1}{2\sqrt{y + x}}\left( \frac{dy}{dx} + 1 \right) + \frac{1}{2\sqrt{y - x}}\left( \frac{dy}{dx} - 1 \right) = 0\]

\[ \Rightarrow \frac{dy}{dx}\left( \frac{1}{2\sqrt{y + x}} \right) + \frac{dy}{dx}\left( \frac{1}{2\sqrt{y - x}} \right) = \frac{1}{2\sqrt{y - x}} - \frac{1}{2\sqrt{y + x}}\]

\[ \Rightarrow \frac{dy}{dx} \times \frac{1}{2}\left[ \frac{1}{\sqrt{y + x}} + \frac{1}{\sqrt{y - x}} \right] = \frac{1}{2}\left[ \frac{\sqrt{y + x} - \sqrt{y - x}}{\sqrt{y - x}\sqrt{y + x}} \right]\]

\[ \Rightarrow \frac{dy}{dx}\left[ \frac{\sqrt{y - x} + \sqrt{y + x}}{\sqrt{y + x}\sqrt{y - x}} \right] = \left[ \frac{\sqrt{y + x} - \sqrt{y - x}}{\sqrt{y - x}\sqrt{y + x}} \right]\]

\[ \Rightarrow \frac{dy}{dx} = \frac{\sqrt{y + x} - \sqrt{y - x}}{\sqrt{y + x} + \sqrt{y - x}} \times \frac{\sqrt{y + x} - \sqrt{y - x}}{\sqrt{y + x} - \sqrt{y - x}} \left[ \text{ rationalizing the denominator } \right]\]

\[ \Rightarrow \frac{dy}{dx} = \frac{\left( y + x \right) + \left( y - x \right) - 2\sqrt{y + x}\sqrt{y - x}}{y + x - y + x}\]

\[ \Rightarrow \frac{dy}{dx} = \frac{2y - 2\sqrt{y^2 - x^2}}{2x}\]

\[ \Rightarrow \frac{dy}{dx} = \frac{2y}{2x} - \frac{2\sqrt{y^2 - x^2}}{2x}\]

\[ \Rightarrow \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2 - x^2}{x^2}}\]

\[ \Rightarrow \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\]

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Solution If √ Y + X + √ Y − X = C , Show that D Y D X = Y X − √ Y 2 X 2 − 1 ? Concept: Simple Problems on Applications of Derivatives.
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