<p>If \(x^2 + y^2 + \sin y = 4\), then the value of \(\dfrac{d^2y}{dx^2}\) at the point \((-2, 0)\) is</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation twice to find d²y/dx², then substitute the point to evaluate. The first derivative gives dy/dx in terms of x and y, and differentiating again yields d²y/dx² which depends on (dy/dx)² and d²y/dx² itself—requiring algebraic solution.
<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: Find dy/dx at (-2, 0)</strong></p><p>dy/dx|₍₋₂,₀₎ = -2(-2)/(2(0) + cos 0) = 4/(0 + 1) = 4</p><p><strong>Step 3: Second implicit differentiation</strong></p><p>Differentiate 2x + 2y(dy/dx) + cos y · (dy/dx) = 0 with respect to x:</p><p>2 + 2(dy/dx)² + 2y(d²y/dx²) - sin y · (dy/dx)² + cos y · (d²y/dx²) = 0</p><p>2 + 2(dy/dx)² - sin y · (dy/dx)² + (2y + cos y)(d²y/dx²) = 0</p><p><strong>Step 4: Substitute point (-2, 0) and dy/dx = 4</strong></p><p>2 + 2(4)² - sin(0) · (4)² + (2(0) + cos 0)(d²y/dx²) = 0</p><p>2 + 32 - 0 + 1 · (d²y/dx²) = 0</p><p>34 + d²y/dx² = 0</p><p>∴ d²y/dx² = -34</p>
Correct Answer: A