Question Detail

We use the quotient rule: d/dx (u/v) = (u'v - uv') / v2.Let u = esin 2x and v = cos 2x.First, find u':u' = d/dx (esin 2x) = esin 2x * d/dx (sin 2x) = esin 2x * (cos 2x * 2) = 2 cos 2x esin 2x.Next, find v':v' = d/dx (cos 2x) = -sin 2x * d/dx (2x) = -2 sin 2x.Now, apply the quotient rule:dy/dx = ( (2 cos 2x esin 2x)(cos 2x) - (esin 2x)(-2 sin 2x) ) / (cos 2x)2dy/dx = ( 2 cos2 2x esin 2x + 2 sin 2x esin 2x ) / cos2 2xFactor out 2esin 2x from the numerator:dy/dx = 2esin 2x (cos2 2x + sin 2x) / cos2 2xSeparate the terms in the numerator over the common denominator: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)Therefore, the correct option is C.