Question Detail
Question
Differentiate the implicit function x2 sin y - e2y cos x = 3x2 + 2x - 5 with respect to x.
Correct Answer
Option C is the correct answer.
Detailed Explanation
Differentiate each term with respect to x using the product rule and chain rule where necessary:1. d/dx (x2 sin y) = 2x sin y + x2 cos y (dy/dx)2. d/dx (-e2y cos x) = -(e2y (-sin x) + cos x * e2y * 2 (dy/dx)) = e2y sin x - 2e2y cos x (dy/dx)3. d/dx (3x2 + 2x - 5) = 6x + 2Combine these:2x sin y + x2 cos y (dy/dx) + e2y sin x - 2e2y cos x (dy/dx) = 6x + 2Group terms with dy/dx:(dy/dx) (x2 cos y - 2e2y cos x) = 6x + 2 - 2x sin y - e2y sin xFinally, solve for dy/dx:dy/dx = (6x + 2 - 2x sin y - e2y sin x) / (x2 cos y - 2e2y cos x)This matches option C.
Hint
Remember to use the product rule for terms like x2 sin y and e2y cos x. Also, apply the chain rule for derivatives of y (e.g., d/dx sin y = cos y dy/dx) and for terms involving e2y (e.g., d/dx e2y = 2e2y dy/dx).