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Solution for If Y = 1 + α ( 1 X − α ) + β / X ( 1 X − α ) ( 1 X − β ) + γ / X 2 ( 1 X − α ) ( 1 X − β ) ( 1 X − γ ) , Find D Y D X ? - CBSE (Science) Class 12 - Mathematics

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Question

\[\text{ If y } = 1 + \frac{\alpha}{\left( \frac{1}{x} - \alpha \right)} + \frac{{\beta}/{x}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)} + \frac{{\gamma}/{x^2}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)\left( \frac{1}{x} - \gamma \right)}, \text{ find } \frac{dy}{dx} \] ?

Solution

\[y = 1 + \frac{\alpha}{\left( \frac{1}{x} - \alpha \right)} + \frac{{\beta}/{x}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)} + \frac{{\gamma}{x^2}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)\left( \frac{1}{x} - \gamma \right)}\]

\[ \Rightarrow y = 1 + \frac{ax}{1 - ax} + \frac{\beta x}{\left( 1 - \ ax \right)\left( 1 - \beta x \right)} + \frac{\gamma x}{\left( 1 - \alpha x \right)\left( 1 - \beta x \right)\left( 1 - \gamma \right)}\]

\[\text{ Taking log on both sides, we get }\]

\[\log y = \log1 + \log\alpha x - \log\left( 1 - \alpha x \right) + \log\beta x - \log\left( 1 - \alpha x \right) - \log\left( 1 - \beta x \right) + \log\gamma x - \log\left( 1 - \alpha x \right) - \log\left( 1 - \beta x \right) - \log\left( 1 - \gamma x \right)\]

\[ \Rightarrow \log y = \log\alpha x - 3\log\left( 1 - \alpha x \right) + \log\beta x - 2\log\left( 1 - \beta x \right) + \log\gamma x - \log\left( 1 - \gamma x \right)\]

\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{1}{x} + \frac{3\alpha}{1 - \alpha x} + \frac{1}{x} + \frac{2\beta}{1 - \beta x} + \frac{1}{x} + \frac{\gamma}{1 - \gamma x}\]

\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{1}{x}\left[ 3 + \frac{3\alpha}{\left( \frac{1}{x} - \alpha \right)} + \frac{2\beta}{\left( \frac{1}{x} - \beta \right)} + \frac{\gamma}{\left( \frac{1}{x} - \gamma \right)} \right]\]

\[ \Rightarrow \frac{dy}{dx} = \frac{y}{x}\left[ \frac{\alpha}{\left( \frac{1}{x} - \alpha \right)} + \frac{\beta}{\left( \frac{1}{x} - \beta \right)} + \frac{\gamma}{\left( \frac{1}{x} - \gamma \right)} \right]\]

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Solution for question: If Y = 1 + α ( 1 X − α ) + β / X ( 1 X − α ) ( 1 X − β ) + γ / X 2 ( 1 X − α ) ( 1 X − β ) ( 1 X − γ ) , Find D Y D X ? concept: Simple Problems on Applications of Derivatives. For the courses CBSE (Science), PUC Karnataka Science, CBSE (Arts), CBSE (Commerce)
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