Question

If \( x + y = xy \), then \( \frac{dy}{dx} \) is

• A. \( \frac{y}{x - 1} \)
• B. \( \frac{1}{x - 1} \)
• C. \( \frac{y - 1}{x - 1} \)
• D. \( \frac{1 - y}{x - 1} \)

Hint:

When an equation involves products of \( x \) and \( y \), always apply the product rule carefully.
Alternatively, you can solve for \( y \) first: \( y(x - 1) = x \implies y = \frac{x}{x - 1} \), then use the quotient rule.

Answer 1

The Correct Option is D

Concept:
• Implicit differentiation: Differentiating both sides of an equation with respect to \( x \) where \( y \) is treated as a function of \( x \).
• Product rule of differentiation: \( \frac{d}{dx}[u(x)v(x)] = u \frac{dv}{dx} + v \frac{du}{dx} \).

Step 1: Differentiate both sides of the equation with respect to \( x \)
The given equation is \( x + y = xy \).
Differentiating with respect to \( x \):
\[ \frac{d}{dx}(x) + \frac{d}{dx}(y) = \frac{d}{dx}(xy) \]
Applying the power rule and product rule:
\[ 1 + \frac{dy}{dx} = x \frac{dy}{dx} + y(1) \]

Step 2: Isolate the terms containing \( \frac{dy}{dx} \)
Rearranging the equation to group all \( \frac{dy}{dx} \) terms on one side:
\[ 1 - y = x \frac{dy}{dx} - \frac{dy}{dx} \]

Step 3: Solve for \( \frac{dy}{dx} \)
Factor out \( \frac{dy}{dx} \) on the right-hand side:
\[ 1 - y = \frac{dy}{dx}(x - 1) \]
\[ \frac{dy}{dx} = \frac{1 - y}{x - 1} \]
This matches option (D).