Question Detail

Use the quotient rule: d/dx (u/v) = (u'v - uv') / v2.Let u = esin 2x and v = cos 2x.Then u' = d/dx [esin 2x] = esin 2x * (cos 2x) * 2 = 2esin 2x cos 2x.And v' = d/dx [cos 2x] = -sin 2x * 2 = -2sin 2x.Now, substitute these into the quotient rule formula:dy/dx = ( (2esin 2x cos 2x) * (cos 2x) - (esin 2x) * (-2sin 2x) ) / (cos 2x)2dy/dx = ( 2esin 2x cos2 2x + 2esin 2x sin 2x ) / cos2 2xFactor out 2esin 2x from the numerator:dy/dx = 2esin 2x ( cos2 2x + sin 2x ) / cos2 2xSeparate the terms in the numerator:dy/dx = 2esin 2x ( cos2 2x / cos2 2x + sin 2x / cos2 2x )dy/dx = 2esin 2x ( 1 + (sin 2x / cos 2x) * (1 / cos 2x) )dy/dx = 2esin 2x ( 1 + tan 2x * sec 2x ).