Question Detail

Rewrite the function as y = esin2x · sec(2x) and use the product rule (uv)' = u'v + uv'.Let u = esin2x, then u' = esin2x · (d/dx(sin2x)) = esin2x · (cos2x · 2) = 2cos2x · esin2x.Let v = sec(2x), then v' = sec(2x)tan(2x) · (d/dx(2x)) = 2sec(2x)tan(2x).Apply the product rule:dy/dx = (2cos2x · esin2x) · sec(2x) + esin2x · (2sec(2x)tan(2x))Since sec(2x) = 1/cos(2x), the first term simplifies:dy/dx = 2cos2x · esin2x · (1/cos2x) + 2esin2xsec(2x)tan(2x)dy/dx = 2esin2x + 2esin2xsec(2x)tan(2x)Factor out 2esin2x:dy/dx = 2esin2x (1 + sec2x tan2x).