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

justaguide | Certified Educator

We have to find the derivative of g(x) =(1+2 sin x) e^(x/2) + cos x

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

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

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

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

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

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

giorgiana1976 | Student

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).

pagu | Student

Find the derivative of g(x) =(1+2 sin x) e^(x/2) + cos x

Use

d(sin u) =cosu du;             d(cos u) = -sinu du

d(e^u) =e^u du;               d(uv) = udv + vdu

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

g'(x) =(1/2+ sin x + 2cosx)e^(x/2) - sin x....ans