Question

If \(\displaystyle \sin\!\left(\frac{x+y}{x-y}\right) = \tan \frac{\pi}{5}\), then find \(\displaystyle \frac{dy}{dx}\).

• A. \( \dfrac{x}{y} \)
• B. \( \dfrac{y}{x} \)
• C. \( -\dfrac{y}{x} \)
• D. \( -\dfrac{x}{y} \)

Hint:

If a trigonometric function equals a constant, first reduce the expression inside the function to a constant before differentiating.

Answer 1

The Correct Option is B

Step 1: Observe the given equation.
Since \(\tan \frac{\pi}{5}\) is a constant, \[ \sin\!\left(\frac{x+y}{x-y}\right) = \text{constant} \] implies \[ \frac{x+y}{x-y} = \text{constant} \]
Step 2: Differentiate implicitly.
Let \[ \frac{x+y}{x-y} = c \] Differentiating both sides with respect to \(x\): \[ \frac{(x-y)(1 + \frac{dy}{dx}) - (x+y)(1 - \frac{dy}{dx})}{(x-y)^2} = 0 \]
Step 3: Simplify the expression.
\[ (x-y)\left(1 + \frac{dy}{dx}\right) = (x+y)\left(1 - \frac{dy}{dx}\right) \]
Step 4: Solve for \(\frac{dy}{dx}\).
\[ \frac{dy}{dx} = \frac{y}{x} \]