Limits, Continuity & Differentiability
Second order derivatives using implicit differentiation
Grade 12
<p>Given: \(x^2 + y^2 + \sin y = 4\). Find \(-\dfrac{d^2y}{dx^2}\bigg|_{(-2,0)}\).</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation twice, then substitute the point (-2,0). At this point, first find dy/dx from the first differentiation, then differentiate again using the chain rule and quotient rule carefully.
<p><strong>Step 1: First implicit differentiation</strong></p><p>Differentiate x² + y² + sin y = 4 with respect to x:</p><p>2x + 2y(dy/dx) + cos y · (dy/dx) = 0</p><p>(2y + cos y)(dy/dx) = -2x</p><p>dy/dx = -2x/(2y + cos y)</p><p><strong>Step 2: Evaluate dy/dx at (-2,0)</strong></p><p>dy/dx|₍₋₂,₀₎ = -2(-2)/(2(0) + cos 0) = 4/1 = 4</p><p><strong>Step 3: Second implicit differentiation</strong></p><p>Differentiate (2y + cos y)(dy/dx) = -2x with respect to x:</p><p>(2y + cos y)(d²y/dx²) + (2 - sin y)(dy/dx)² = -2</p><p><strong>Step 4: Substitute at (-2,0)</strong></p><p>(2(0) + cos 0)(d²y/dx²) + (2 - sin 0)(4)² = -2</p><p>(1)(d²y/dx²) + (2)(16) = -2</p><p>d²y/dx² + 32 = -2</p><p>d²y/dx² = -34</p><p><strong>Step 5: Find -d²y/dx²</strong></p><p>-d²y/dx²|₍₋₂,₀₎ = -(-34) = 34</p><p>∴ Answer: <strong>34</strong></p>
Correct Answer: B