Question

If  \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?

 

Answer 1

\[\text{ We have }, xy = e^\left( x - y \right)\]

Taking log on both sides,

\[\log\left( xy \right) = \log\left( e^{x - y} \right)\]
\[ \Rightarrow \log x + \log y = \left( x - y \right)\log e\]
\[ \Rightarrow \log x + \log y = \left( x - y \right) \times 1\]
\[ \Rightarrow \log x + \log y = x - y\]

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