Question

If $$ \sin x \sqrt{\cos y} - \cos y \sqrt{\sin x} = 0, $$ then find $$ \frac{dy}{dx}. $$

• A. \( \tan x \)
• B. 1
• C. -1
• D. \( -\cot x \)

Hint:

Use implicit differentiation carefully on composite functions.

Answer 1

The Correct Option is C

Differentiate implicitly w.r.t \( x \): \[ \frac{d}{dx} \left( \sin x \sqrt{\cos y} \right) - \frac{d}{dx} \left( \cos y \sqrt{\sin x} \right) = 0 \] Calculate derivatives: \[ \cos x \sqrt{\cos y} + \sin x \cdot \frac{1}{2} (\cos y)^{-1/2} (-\sin y) \frac{dy}{dx} - (-\sin y) \sqrt{\sin x} \frac{dy}{dx} - \cos y \cdot \frac{1}{2} (\sin x)^{-1/2} \cos x = 0 \] Simplify and solve for \( \frac{dy}{dx} \), result is: \[ \frac{dy}{dx} = -1 \]