Use the product rule to find the derivative of g(x)=(1+2 sin x) e^(x/2) + cos x

The product rule must be applied to the first term of the
sum only, if the expression of the function is g(x) = (1+2 sin x) e^(x/2) + cos
x.

If the expression of the function is g(x) = (1+2 sin
x)(e^(x/2) + cos x), then the product rule will be applied as it
follows:

g'(x) = (1+2 sin x)'*(e^(x/2) + cos x) + (1+2 sin
x)*(e^(x/2) + cos x)'

g'(x) = 2cos x*(e^(x/2) + cos x) +
(1+2 sin x)*((e^(x/2))/2 - sin x)

Therefore,
the requested derivative of the function g(x)= (1+2 sin x)(e^(x/2) + cos x) is g'(x)
=  2cos x*(e^(x/2) + cos x) + (1+2 sin x)*((e^(x/2))/2 - sin
x).