Verified answer

The proposition is ALMOST self-evident...

Without assuming that f(x) - g(x) is a constant, just give the name h(x) to the quantity f(x)-g(x):

h(x) = f(x) - g(x).

Now take the derivative, with respect to x, of both sides:

h'(x) = f '(x) - g'(x)

But the problem statement assures us that the right-hand side is zero.

Thus, h'(x) is zero everywhere, meaning that it does not change when x changes,

meaning that it must be constant.