When differentiating an implicit equation, always ensure to apply the chain rule correctly. In cases where the equation involves trigonometric functions, be sure to use their derivative identities. Additionally, when isolating \(\frac{dy}{dx}\), carefully rearrange terms to simplify the expression. In this problem, substituting the expression for \(x\) into the denominator helped simplify the final result.
The provided equation is:
\(\sin y = x \sin(a + y)\).
Differentiating both sides with respect to \(x\) yields:
\(\cos y \frac{dy}{dx} = \sin(a + y) + x \cos(a + y) \frac{dy}{dx}\).
Rearranging to isolate \(\frac{dy}{dx}\) gives:
\(\frac{dy}{dx} (\cos y - x \cos(a + y)) = \sin(a + y)\).
Simplifying the expression for \(\frac{dy}{dx}\):
\(\frac{dy}{dx} = \frac{\sin(a + y)}{\cos y - x \cos(a + y)}\).
From the initial equation \(\sin y = x \sin(a + y)\), we express \(x\) as:
\(x = \frac{\sin y}{\sin(a + y)}\).
Substituting this expression for \(x\) into the denominator:
\(\cos y - x \cos(a + y) = \cos y - \frac{\sin y \cos(a + y)}{\sin(a + y)}\).
Simplifying the denominator further:
\(\cos y - x \cos(a + y) = \frac{\cos y \sin(a + y) - \sin y \cos(a + y)}{\sin(a + y)} = \frac{\sin a}{\sin(a + y)}\).
Substituting this simplified denominator back into the expression for \(\frac{dy}{dx}\):
\(\frac{dy}{dx} = \frac{\sin(a + y)}{\frac{\sin a}{\sin(a + y)}} = \frac{\sin^2(a + y)}{\sin a}\).
Therefore, the final result is:
\(\boxed{\frac{\sin^2(a + y)}{\sin a}}\).