Question

If \( \sqrt{\frac{y}{x}} + 4\sqrt{\frac{x}{y}} = 4 \), then \( \frac{dy}{dx} \):

• A. \( xy \)
• B. \( \frac{x}{y} \)
• C. \( -4 \)
• D. \( 4 \)

Hint:

When solving equations with square roots, square both sides and simplify step by step. Use differentiation techniques like the product rule and chain rule to find the derivative accurately.

Answer 1

The Correct Option is D

The given equation is: \[ \sqrt{\frac{y}{x}} + 4\sqrt{\frac{x}{y}} = 4. \] Step 1: Square both sides.
Squaring both sides yields: \[ \left(\sqrt{\frac{y}{x}} + 4\sqrt{\frac{x}{y}}\right)^2 = 4^2. \] Expanding the left side: \[ \frac{y}{x} + 16 \cdot \frac{x}{y} + 2 \cdot 4 \cdot \sqrt{\frac{y}{x} \cdot \frac{x}{y}} = 16. \] Simplifying: \[ \frac{y}{x} + 16 \cdot \frac{x}{y} + 8 = 16. \] Rearranging the terms: \[ \frac{y}{x} + 16 \cdot \frac{x}{y} = 8. \] Step 2: Eliminate fractions.
Multiply by \( xy \) to clear denominators: \[ y^2 + 16x^2 = 8xy. \] Step 3: Differentiate with respect to \( x \).
Differentiating both sides with respect to \( x \): \[ \frac{d}{dx}(y^2) + \frac{d}{dx}(16x^2) = \frac{d}{dx}(8xy). \] Applying the chain rule: \[ 2y \frac{dy}{dx} + 32x = 8\left(y + x \frac{dy}{dx}\right). \] Simplifying: \[ 2y \frac{dy}{dx} + 32x = 8y + 8x \frac{dy}{dx}. \] Isolating \( \frac{dy}{dx} \) terms: \[ 2y \frac{dy}{dx} - 8x \frac{dy}{dx} = 8y - 32x. \] Factoring out \( \frac{dy}{dx} \): \[ (2y - 8x) \frac{dy}{dx} = 8y - 32x. \] Solving for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{8y - 32x}{2y - 8x}. \] Step 4: Simplify the expression for \( \frac{dy}{dx} \).
Factoring the numerator and denominator: \[ \frac{dy}{dx} = \frac{8(y - 4x)}{2(y - 4x)}. \] Canceling the common factor \( (y - 4x) \): \[ \frac{dy}{dx} = 4. \] Final Answer: \[ \boxed{4} \]