Question Detail

Differentiate each term with respect to x using implicit differentiation.1. d/dx(x2sin y): Using the product rule ((uv)'=u'v+uv') where u=x2, v=sin y. d/dx(x2sin y) = (2x)sin y + x2(cos y)(dy/dx)2. d/dx(e2ycos x): Using the product rule where u=e2y, v=cos x. d/dx(e2ycos x) = (e2y)(2)(dy/dx)cos x + e2y(-sin x) = 2e2ycos x (dy/dx) - e2ysin x3. d/dx(3x2 + 2x - 5) = 6x + 2Substitute these back into the original equation:2x sin y + x2 cos y (dy/dx) - [2e2y cos x (dy/dx) - e2y sin x] = 6x + 22x sin y + x2 cos y (dy/dx) - 2e2y cos x (dy/dx) + e2y sin x = 6x + 2Group terms with dy/dx on one side and other terms on the other:dy/dx (x2 cos y - 2e2y cos x) = 6x + 2 - 2x sin y - e2y sin xSolve for dy/dx:dy/dx = (6x + 2 - 2x sin y - e2y sin x) / (x2 cos y - 2e2y cos x)Rearrange the numerator terms and rewrite e2ysin x as sin x e2y and 2e2ycos x as 2cos x e2y for clarity:dy/dx = (6x - 2x sin y - sin x e2y + 2) / (x2 cos y - 2cos x e2y)This matches option C.