Question Detail

This problem requires the quotient rule: (u/v)' = (u'v - uv') / v2.Let u = esin 2x and v = cos 2x.First, find the derivatives of u and v using the chain rule:u' = d/dx(esin 2x) = esin 2x * d/dx(sin 2x) = esin 2x * (cos 2x) * 2 = 2cos 2x esin 2xv' = d/dx(cos 2x) = -sin 2x * d/dx(2x) = -2sin 2xNow apply the quotient rule:dy/dx = ( (2cos 2x esin 2x)(cos 2x) - (esin 2x)(-2sin 2x) ) / (cos 2x)2dy/dx = ( 2cos2 2x esin 2x + 2sin 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: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)This matches option C.