Question

If \( x^x y^y = e^e \), then \( \left( \frac{d^2 y}{dx^2} \right)_{(e, e)} = \):

• A. \( \frac{1}{e} \left( \frac{dy}{dx} \right)_{(e,e)} \)
• B. \( \left( \frac{dy}{dx} \right)_{(e,e)} + \frac{1}{e} \)
• C. \( \left( \frac{dy}{dx} \right)_{(e,e)} - \frac{1}{e} \)
• D. \( e \left( \frac{dy}{dx} \right)_{(e,e)} \)

Hint:

For implicit differentiation, remember to apply the chain rule carefully and solve step by step for higher derivatives.

Answer 1

The Correct Option is A

We are given that \( x^x y^y = e^e \). To find \( \left( \frac{d^2 y}{dx^2} \right)_{(e,e)} \), we first differentiate the equation \( x^x y^y = e^e \) with respect to \( x \) using implicit differentiation. After performing the first and second differentiation, we find that the second derivative is: \[ \left( \frac{d^2 y}{dx^2} \right)_{(e,e)} = \frac{1}{e} \left( \frac{dy}{dx} \right)_{(e,e)} \] Thus, the correct answer is option (1).