AP DSC School Assistant Math Model Test 3

Concept:We use implicit differentiation (differentiate both sides with respect to $x$, treating $y$ as a function of $x$) and then simplify using the given relation.Explanation:Given: $\sin y = x \sin(a+y)$.Differentiate both sides with respect to $x$:$\frac{d}{dx}(\sin y) = \frac{d}{dx}[x \sin(a+y)]$Left side: $\cos y \cdot \frac{dy}{dx}$Right side: by product rule, $1\cdot \sin(a+y) + x \cdot \cos(a+y) \cdot \frac{dy}{dx}$So we have:$\cos y \frac{dy}{dx} = \sin(a+y) + x\cos(a+y)\frac{dy}{dx}$Bring the terms with $\frac{dy}{dx}$ to one side:$\cos y \frac{dy}{dx} - x\cos(a+y)\frac{dy}{dx} = \sin(a+y)$Factor out $\frac{dy}{dx}$:$(\cos y - x\cos(a+y))\frac{dy}{dx} = \sin(a+y)$Solve for $\frac{dy}{dx}$:$\frac{dy}{dx} = \frac{\sin(a+y)}{\cos y - x\cos(a+y)}$Now substitute $x = \frac{\sin y}{\sin(a+y)}$ from the original equation into the denominator:$\cos y - \frac{\sin y}{\sin(a+y)}\cos(a+y) = \frac{\cos y\sin(a+y) - \sin y\cos(a+y)}{\sin(a+y)} = \frac{\sin\big((a+y)-y\big)}{\sin(a+y)} = \frac{\sin a}{\sin(a+y)}$Hence,$\frac{dy}{dx} = \frac{\sin(a+y)}{\frac{\sin a}{\sin(a+y)}} = \frac{\sin^{2}(a+y)}{\sin a}$Answer:Option D: $\frac{\sin^{2}(a+y)}{\sin a}$