Question

If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?

Answer 1

\[\text{ We have,} x^m y^n = 1\]

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,

\[\frac{dy}{dx}\left( m \log x \right) + \frac{d}{dx}\left( n \log y \right) = \frac{d}{dx}\left\{ \log\left( 1 \right) \right\}\]

\[ \Rightarrow \frac{m}{x} + \frac{n}{y}\frac{dy}{dx} = 0\]

\[ \Rightarrow \frac{dy}{dx} = - \frac{m}{x} \times \frac{y}{n}\]

\[ \Rightarrow \frac{dy}{dx} = - \frac{my}{nx}\]