Find all functions f: R → R that satisfy the following conditions:
- f(x) is a continuous and differentiable function throughout the domain of real numbers.
- f(0) = 1 and f(1) = e (where e is the base of the natural logarithm).
- f(x)f(y) = f(xy) + f(x + y) for all real numbers x and y.
- f'(x) = f(x) for all real numbers x.