Question

\[\text{ If } x = e^{x/y} , \text{ prove that } \frac{dy}{dx} = \frac{x - y}{x\log x}\] ?

Answer 1

\[x = e^\frac{x} {y} \]
\[\text{ Taking logarithm on both sides, we get  }\]
\[\log x = \frac{x}{y}\]
\[ \Rightarrow y\log x = x\]
\[ \Rightarrow \log x\frac{dy}{dx} + \frac{y}{x} = 1\]
\[ \Rightarrow \log x\frac{dy}{dx} = 1 - \frac{y}{x}\]
\[ \Rightarrow \log x\frac{dy}{dx} = \frac{x - y}{x}\]
\[ \Rightarrow x\log x\frac{dy}{dx} = x - y\]
\[ \Rightarrow \frac{dy}{dx} = \frac{x - y}{x\log x}\]