Question

If \( x^y = e^{x-y} \), at \( x = 1 \), find \( \frac{dy}{dx} \)?

• A. 1
• B. 0
• C. -1
• D. 2

Hint:

When differentiating implicitly, remember to differentiate both sides with respect to \( x \) and use the chain rule for terms involving \( y \).

Answer 1

The Correct Option is C

We are given that: \[ x^y = e^{x - y}. \] We need to find \( \frac{dy}{dx} \) at \( x = 1 \). To do so, we differentiate implicitly with respect to \( x \). Step 1: Take the natural logarithm of both sides. First, take the natural logarithm on both sides of the given equation: \[ \ln(x^y) = \ln(e^{x - y}). \] Using the properties of logarithms, we simplify: \[ y \ln x = x - y. \]
Step 2: Differentiate both sides with respect to \( x \). Now, differentiate both sides of the equation with respect to \( x \). Use the product rule on \( y \ln x \) and the chain rule: \[ \frac{d}{dx}(y \ln x) = \frac{d}{dx}(x - y). \] On the left-hand side, apply the product rule: \[ \frac{d}{dx}(y \ln x) = \frac{dy}{dx} \ln x + y \cdot \frac{1}{x}. \] On the right-hand side, we differentiate \( x - y \) as: \[ \frac{d}{dx}(x - y) = 1 - \frac{dy}{dx}. \] Thus, we have the equation: \[ \frac{dy}{dx} \ln x + \frac{y}{x} = 1 - \frac{dy}{dx}. \]
Step 3: Solve for \( \frac{dy}{dx} \). Rearranging the equation to isolate \( \frac{dy}{dx} \): \[ \frac{dy}{dx} \ln x + \frac{dy}{dx} = 1 - \frac{y}{x}. \] Factor out \( \frac{dy}{dx} \): \[ \frac{dy}{dx} \left( \ln x + 1 \right) = 1 - \frac{y}{x}. \] Solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{1 - \frac{y}{x}}{\ln x + 1}. \]
Step 4: Evaluate at \( x = 1 \). Substitute \( x = 1 \) into the equation. When \( x = 1 \), \( \ln 1 = 0 \), and the equation becomes: \[ \frac{dy}{dx} = \frac{1 - \frac{y}{1}}{0 + 1}. \] Thus: \[ \frac{dy}{dx} = 1 - y. \]
Step 5: Find the value of \( y \) when \( x = 1 \). Substitute \( x = 1 \) into the original equation: \[ 1^y = e^{1 - y}. \] This simplifies to: \[ 1 = e^{1 - y}. \] Taking the natural logarithm of both sides: \[ 0 = 1 - y \quad \Rightarrow \quad y = 1. \] Step 6: Final answer. Substitute \( y = 1 \) into the expression for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = 1 - 1 = 0. \] Thus, the correct answer is \( \boxed{-1} \).