find the linearization of f(s)= (u*s)/(1+s); a=1, and u is positive ? Would I have to start this with the quotient rule and what would I do from there?
You can use the quotient rule, but you might note that
s/(1+s) = 1 - 1/(1+s)
so, f(s) = u(1 - 1/(1+s))
f'(s) = u/(1+s)^2