Question about lebesgue space and fourier transforms

Hi, i need some help with this problem. The material is from a course called 'Numerical functional analysis'. if f(x)=u(x)*exp{-alpha*x} for alpha >0. Where u(x) is the step function ie u(x)=0 for x<0 and u(x)=1 for x larger than or equal to 0. Then does f prime (ie 1st derivative of f) belong to lebesgue space with p=2 over the real line.

Earlier parts of the question ask to calculate fourier transform for f and f prime. I am thinking along the lines of plancherel's theorem. This would mean i would have to show f prime is a fourier transform which can be checked using the inverse fourier transform formula to find the original function. However then i would have to prove this function belongs to the intersection of lebesgue space p=1 and lebesgue space p=2 both over real line. (this is part of plancherel's theorem) Any suggestions would be much appreciated thanks.(Nod)