Limits, Continuity & Differentiability
Differentiation (Implicit)
Grade 12
Question:
<p>If \(\sin y = x\sin(a+y)\), then \(\dfrac{dy}{dx}\) equals:</p>
<p>(1) \(\dfrac{\sin^2(a+y)}{\sin a \cdot \sin y}\)</p>
<p>(2) \(\dfrac{\sin^2(a+y)}{\sin a}\)</p>
<p>(3) \(\dfrac{\sin(a+y)}{\sin a}\)</p>
<p>(4) \(\dfrac{\sin^2 a}{\sin^2(a+y)}\)</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation on the equation sin y = x sin(a+y), recognizing that y is a function of x. The product rule on the right side is critical since x is multiplied by sin(a+y).
<p><strong>Step 1:</strong> Differentiate both sides with respect to x:</p><p>cos y · (dy/dx) = sin(a+y) + x · cos(a+y) · (dy/dx)</p><p><strong>Step 2:</strong> Expand using product rule on right side and chain rule on both sides.</p><p><strong>Step 3:</strong> Collect all dy/dx terms on the left:</p><p>cos y · (dy/dx) - x · cos(a+y) · (dy/dx) = sin(a+y)</p><p><strong>Step 4:</strong> Factor out dy/dx:</p><p>(dy/dx)[cos y - x cos(a+y)] = sin(a+y)</p><p><strong>Step 5:</strong> Isolate dy/dx:</p><p>dy/dx = sin(a+y) / [cos y - x cos(a+y)]</p><p><strong>Step 6:</strong> From the original equation sin y = x sin(a+y), we can express the denominator as: cos y - x cos(a+y) = cos y - [sin y / sin(a+y)] · cos(a+y) = [cos y sin(a+y) - sin y cos(a+y)] / sin(a+y) = sin(a+y-y) / sin(a+y) = sin a / sin(a+y)</p><p><strong>Step 7:</strong> Therefore:</p><p>dy/dx = sin(a+y) / [sin a / sin(a+y)] = sin²(a+y) / sin a</p><p>∴ Answer: B</p>
Correct Answer: B