Question

If \( x = e^{\frac{x}{y}} \), then prove that \( \frac{dy}{dx} = \frac{x - y}{x \log x} \).

Hint:

For equations involving logarithms and exponentials, taking the natural logarithm can simplify differentiation.

Answer 1

Step 1: Apply the natural logarithm to both sides.
Starting equation: \[ x = e^{\frac{x}{y}} \]
Taking the natural logarithm: \[ \log x = \frac{x}{y}. \] Step 2: Differentiate implicitly with respect to \( x \).
Differentiating both sides: \[ \frac{d}{dx} (\log x) = \frac{d}{dx} \left( \frac{x}{y} \right). \]
Applying derivative rules: \[ \frac{1}{x} \cdot \frac{dx}{dx} = \frac{y \frac{dx}{dx} - x \frac{dy}{dx}}{y^2}. \]
Since \( \frac{dx}{dx} = 1 \): \[ \frac{1}{x} = \frac{y - x \frac{dy}{dx}}{y^2}. \]
Step 3: Isolate \( \frac{dy}{dx} \).
Multiply both sides by \( y^2 \): \[ \frac{y^2}{x} = y - x \frac{dy}{dx}. \]
Rearrange terms: \[ x \frac{dy}{dx} = y - \frac{y^2}{x}. \]
Divide by \( x \): \[ \frac{dy}{dx} = \frac{y - \frac{y^2}{x}}{x}. \]
Simplify the expression: \[ \frac{dy}{dx} = \frac{xy - y^2}{x^2}. \]
This concludes the derivation.