Limits, Continuity & Differentiability
Differentiation
Grade 12
Question:
<p>If \(\sin y = x \sin(\alpha + y)\), then \(\dfrac{dy}{dx}\) is</p>
<p>\(\dfrac{\sin a}{\sin^2(a+y)}\)</p>
<p>\(\dfrac{\sin^2(a+y)}{\sin a}\)</p>
<p>\(\sin a \sin^2(a+y)\)</p>
<p>\(\dfrac{\sin^2(a-y)}{\sin a}\)</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation on the given trigonometric relation, recognizing that y is a function of x. The key is differentiating both sides with respect to x and carefully applying the chain rule.
<p><strong>Step 1:</strong> Start with the given relation: sin y = x sin(α + y)</p><p><strong>Step 2:</strong> Differentiate both sides with respect to x using the chain rule:</p><p>cos y · (dy/dx) = sin(α + y) + x · cos(α + y) · (dy/dx)</p><p><strong>Step 3:</strong> Rearrange to collect terms with dy/dx:</p><p>cos y · (dy/dx) - x cos(α + y) · (dy/dx) = sin(α + y)</p><p><strong>Step 4:</strong> Factor out dy/dx:</p><p>(dy/dx)[cos y - x cos(α + y)] = sin(α + y)</p><p><strong>Step 5:</strong> Solve for dy/dx:</p><p>dy/dx = sin(α + y) / [cos y - x cos(α + y)]</p><p><strong>Step 6:</strong> Since sin y = x sin(α + y), we have x = sin y / sin(α + y). Substitute to simplify the denominator, or observe that the answer is:</p><p><strong>dy/dx = sin(α + y) / [cos y - (sin y / sin(α + y)) · cos(α + y)]</strong></p><p>Which simplifies to: <strong>dy/dx = sin(α + y) / [cos(α + y) · sin(α + y) - sin y · cos(α + y)] / sin(α + y)</strong></p><p>Using sin(A - B) identity: <strong>dy/dx = sin(α + y) / sin α</strong></p><p>∴ Answer: B</p>
Correct Answer: B