Question

If \[ \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 4, \quad \text{then} \quad \frac{dy}{dx} = \]

• A. \( \frac{y - 7x}{7x - y} \)
• B. \( \frac{7y - x}{y - 7x} \)
• C. \( \frac{7x + y}{x - 7y} \)
• D. \( \frac{y + 7x}{7y - x} \)

Hint:

When differentiating equations with square roots, use implicit differentiation and the chain rule carefully.

Answer 1

The Correct Option is B

Step 1: Differentiating the given equation.
We are given the equation \( \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = 4 \). To differentiate this implicitly, we first rewrite the equation as: \[ \frac{\sqrt{x}}{\sqrt{y}} + \frac{\sqrt{y}}{\sqrt{x}} = 4 \] Now, differentiate both sides with respect to \( x \). Use the chain rule for each term.

Step 2: Simplifying the derivative.
After differentiating, we obtain the expression for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{7y - x}{y - 7x} \]
Step 3: Conclusion.
Thus, the value of \( \frac{dy}{dx} \) is \( \frac{7y - x}{y - 7x} \), which makes option (B) the correct answer.