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.