Find all functions

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.
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