Question Detail

This problem requires the quotient rule: d/dx(u/v) = (u'v - uv')/v2.Let u = esin2x and v = cos2x.Calculate u': u' = d/dx(esin2x) = esin2x * d/dx(sin2x) = esin2x * cos2x * 2 = 2cos2x esin2x.Calculate v': v' = d/dx(cos2x) = -sin2x * 2 = -2sin2x.Now apply the quotient rule:dy/dx = ( (2cos2x esin2x) * cos2x - esin2x * (-2sin2x) ) / (cos2x)2dy/dx = ( 2cos22x esin2x + 2sin2x esin2x ) / cos22xFactor out 2esin2x from the numerator:dy/dx = 2esin2x ( cos22x + sin2x ) / cos22xSeparate the terms:dy/dx = 2esin2x ( cos22x / cos22x + sin2x / cos22x )dy/dx = 2esin2x ( 1 + (sin2x/cos2x) * (1/cos2x) )dy/dx = 2esin2x ( 1 + tan2x sec2x ).Note: The options show 2esinx instead of 2esin2x. Assuming this is a typo in the options and it should be sin2x, option C matches the form.