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.

Find all functions f: R → R that satisfy the following conditions:

1. f(x) is a continuous and differentiable function throughout the domain of real numbers.
2. f(0) = 1 and f(1) = e (where e is the base of the natural logarithm).
3. f(x)f(y) = f(xy) + f(x + y) for all real numbers x and y.
4. f'(x) = f(x) for all real numbers x.