Question Detail

We can rewrite the function as a product: y = esin2x (cos2x)-1.Now apply the product rule: d/dx(uv) = u'v + uv'.Let u = esin2x. Then u' = d/dx(esin2x) = esin2x * (d/dx(sin2x)) = esin2x * (cos2x * 2) = 2cos2x esin2x.Let v = (cos2x)-1. Then v' = d/dx((cos2x)-1) = -1(cos2x)-2 * (d/dx(cos2x))= -(cos2x)-2 * (-sin2x * 2) = 2sin2x / (cos2x)2.Substitute into the product rule:dy/dx = (2cos2x esin2x)(cos2x)-1 + (esin2x)(2sin2x / (cos2x)2)dy/dx = 2esin2x + esin2x (2sin2x / (cos2x)2)Factor out 2esin2x:dy/dx = 2esin2x [1 + sin2x / (cos2x)2]Rewrite the term inside the bracket:sin2x / (cos2x)2 = (sin2x / cos2x) * (1 / cos2x) = tan2x * sec2x.So, dy/dx = 2esin2x [1 + tan2x sec2x].This matches option C.