Question Detail
Question
Differentiate the implicit functions x2 siny - e2y cosx = 3x2 + 2x - 5 with respect to x.
Correct Answer
Option C is the correct answer.
Detailed Explanation
Differentiate each term with respect to x:1. d/dx(x2 siny) = 2x siny + x2 cos y (dy/dx) (using product rule and chain rule)2. d/dx(e2y cosx) = (d/dx(e2y)) cosx + e2y (d/dx(cosx)) = (2e2y dy/dx) cosx + e2y (-sin x) = 2e2y cosx (dy/dx) - e2y sinx (using product rule and chain rule)3. d/dx(3x2 + 2x - 5) = 6x + 2Substitute these back into the original equation:(2x siny + x2 cos y dy/dx) - (2e2y cosx dy/dx - e2y sinx) = 6x + 22x siny + x2 cos y dy/dx - 2e2y cosx dy/dx + e2y sinx = 6x + 2Group terms with dy/dx:dy/dx (x2 cos y - 2e2y cosx) = 6x + 2 - 2x siny - e2y sinxSolve for dy/dx:dy/dx = (6x + 2 - 2x siny - e2y sinx)⁄(x2 cos y - 2e2y cosx)This matches option C if we reorder the numerator terms and recognize sinxe2y as e2ysinx.
Hint
Use implicit differentiation, applying the product rule and chain rule carefully for all terms. Remember that d/dx(ef(y)) = ef(y) * f'(y) * dy/dx.