Question Detail

To differentiate the implicit equation x² sin y - e^(2y) cos x = 3x² + 2x - 5 with respect to x, we apply implicit differentiation.1. Differentiate x² sin y using the product rule: d/dx(x² sin y) = (d/dx(x²)) sin y + x² (d/dx(sin y)) = 2x sin y + x² cos y (dy/dx)2. Differentiate -e^(2y) cos x using the product rule and chain rule: d/dx(-e^(2y) cos x) = -( (d/dx(e^(2y))) cos x + e^(2y) (d/dx(cos x)) ) = -( (2e^(2y) dy/dx) cos x + e^(2y) (-sin x) ) = -2e^(2y) cos x (dy/dx) + e^(2y) sin x3. Differentiate the right side: d/dx(3x² + 2x - 5) = 6x + 2Now, combine these derivatives:2x sin y + x² cos y (dy/dx) - 2e^(2y) cos x (dy/dx) + e^(2y) sin x = 6x + 2Isolate terms with dy/dx:(dy/dx) (x² cos y - 2e^(2y) cos x) = 6x + 2 - 2x sin y - e^(2y) sin xSolve for dy/dx:dy/dx = (6x - 2x sin y - e^(2y) sin x + 2) / (x² cos y - 2e^(2y) cos x)Comparing this result to the given options, the numerator matches that of option A. However, the denominator in option A is x² cos y + 2 cos x e^(2y), which has a positive sign before 2cosxe^(2y) instead of a negative sign. This discrepancy suggests a possible typo in the problem statement (e.g., if the original term was +e^(2y)cos x) or in the options. However, as option A is selected in the source, we provide the derivation for the closest matching answer despite the sign discrepancy in the denominator that would arise from the given equation. If the original equation had been x² sin y + e^(2y) cos x = 3x² + 2x - 5, the denominator would match, but the numerator would still have a sign difference for the e^(2y)sin x term. Therefore, the explanation aligns with the structure of the provided option A, acknowledging these potential inconsistencies.