Question

If \(\sqrt{x} + \sqrt{y} = \sqrt{xy}\), then find \(\dfrac{dy}{dx}\).

• A. \(-\left(\dfrac{y}{x}\right)^{\frac{3}{2}}\)
• B. \(\left(\dfrac{x}{y}\right)^{\frac{3}{2}}\)
• C. \(-\left(\dfrac{x}{y}\right)^{\frac{3}{2}}\)
• D. \(\left(\dfrac{y}{x}\right)^{\frac{3}{2}}\)

Hint:

In equations involving square roots, squaring both sides first simplifies differentiation.

Answer 1

The Correct Option is A

Step 1: Square both sides.
\[ \sqrt{x} + \sqrt{y} = \sqrt{xy} \Rightarrow x + y + 2\sqrt{xy} = xy \]
Step 2: Differentiate implicitly w.r.t. \(x\).
\[ 1 + \frac{dy}{dx} + \frac{2}{2\sqrt{xy}}(y + x\frac{dy}{dx}) = y + x\frac{dy}{dx} \]
Step 3: Simplify and solve for \(\dfrac{dy}{dx}\).
After simplification, we get \[ \frac{dy}{dx} = -\left(\frac{y}{x}\right)^{\frac{3}{2}} \]