On Page 78 of Lang - Undergraduate Analysis (2nd Ed) we find the following:

"We assume that there is a function f defined for all numbers such that f ' = f

and f(0) = 1.

We note that f(x)is not equal to 0 for all x. Indeed differentiating the function

f(x)f(-x) we find 0"

Can anyone help me show rigorously that the derivative of f(x)f(-x) for the exponential function is zero - relying only on f ' = f and f(0) = 1.

Help will be appreciated.