Question

If \( x\sqrt{1 + y} + y\sqrt{1 + x} = 0 \), then find \( \frac{dy}{dx} \).

• A. \( x + \frac{1}{x} \)
• B. \( \frac{1}{1 + x} \)
• C. \( -\frac{1}{(1 + x)^2} \)
• D. \( \frac{x}{1 + x} \)

Hint:

When differentiating implicitly, apply the product and chain rule carefully, and isolate \( \frac{dy}{dx} \) to solve for it.

Answer 1

The Correct Option is C

Step 1: {Implicit Differentiation}
Given the equation \( x\sqrt{1 + y} + y\sqrt{1 + x} = 0 \), differentiate both sides with respect to \( x \): \[ \frac{d}{dx} \left(x\sqrt{1 + y}\right) + \frac{d}{dx} \left(y\sqrt{1 + x}\right) = 0 \] Using the product rule: \[ \frac{d}{dx} \left(x\sqrt{1 + y}\right) = \sqrt{1 + y} + x \cdot \frac{1}{2\sqrt{1 + y}} \cdot \frac{dy}{dx} \] \[ \frac{d}{dx} \left(y\sqrt{1 + x}\right) = \sqrt{1 + x} \cdot \frac{dy}{dx} + y \cdot \frac{1}{2\sqrt{1 + x}} \] Now, substitute and solve for \( \frac{dy}{dx} \). 
Step 2: {Solve for \( \frac{dy}{dx} \)}
After simplifying, we find: \[ \frac{dy}{dx} = -\frac{1}{(1 + x)^2} \]