Question

Find \( \frac{dy}{dx} \) where \( a^y = \left( \frac{x}{y} \right)^a \)

• A. \( xy \)
• B. \( \cancel{} \)
• C. \( \frac{x}{y} \)
• D. Does not exist

Hint:

Whenever differentiating equations that involve variables both in the base and exponent, check if simplifying or rewriting the equation helps in solving.

Answer 1

The Correct Option is B

We are given the equation \( a^y = \left( \frac{x}{y} \right)^a \). To differentiate implicitly, we apply logarithms to both sides. However, based on the structure of the equation, solving for \( y \) explicitly may lead to inconsistencies or undefined solutions in this case, implying that no valid differentiation exists under the given conditions.