Question

If $y=\cos x \cos y$, then $\frac{dy}{dx}$ at $\left(\frac{\pi}{3},\frac{\pi}{6}\right)$ is:

• A. $\frac{-3}{5}$
• B. $\frac{3}{5}$
• C. $\frac{-5}{3}$
• D. $\frac{-4}{3}$
• E. $\frac{-8}{3}$

Hint:

Group all terms containing $\frac{dy}{dx}$ on one side of the equation to isolate the derivative.

Answer 1

The Correct Option is A

Step 1: Concept
Use implicit differentiation: differentiate both sides with respect to $x$.

Step 2: Analysis
$\frac{dy}{dx} = \frac{d}{dx}(\cos x \cos y)$. $\frac{dy}{dx} = -\sin x \cos y + \cos x (-\sin y \frac{dy}{dx})$. $\frac{dy}{dx} (1 + \cos x \sin y) = -\sin x \cos y$.

Step 3: Calculation
Substitute $x = \pi/3, y = \pi/6$: $\sin(\pi/3) = \sqrt{3}/2, \cos(\pi/3) = 1/2$. $\sin(\pi/6) = 1/2, \cos(\pi/6) = \sqrt{3}/2$. $\frac{dy}{dx} (1 + \frac{1}{2} \cdot \frac{1}{2}) = -\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}$. $\frac{dy}{dx} (\frac{5}{4}) = -\frac{3}{4} \implies \frac{dy}{dx} = -\frac{3}{5}$. Final Answer: (A)