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Question
If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
Solution
\[\text{ We have,} x^m y^n = 1\]
Taking log on both side,
\[\log\left( x^m y^n \right) = \log\left( 1 \right)\]
\[ \Rightarrow m \log x + n \log y = \log\left( 1 \right)\]
Differentiating with respect to x,
\[\frac{dy}{dx}\left( m \log x \right) + \frac{d}{dx}\left( n \log y \right) = \frac{d}{dx}\left\{ \log\left( 1 \right) \right\}\]
\[ \Rightarrow \frac{m}{x} + \frac{n}{y}\frac{dy}{dx} = 0\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{m}{x} \times \frac{y}{n}\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{my}{nx}\]
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