Gary Scheidegger wrote at 2015-07-22 04:51:56
The rule (e^f(x))' = f'(x) e^f(x) is a special form of the more general rule Dx(g(f(x)) = Dx(f(x))Df(g(x)), where Dx is the derivative with respect to x and Df is the derivative with respect to f, and f and g are both functions. Thus to say (e^f(x))' = f'(x) e^f(x) presupposes that Dx e^x = e^x, or in this case Df e^f(x) = e^f(x).
Indeed one, but by no means only, way of DEFINING e is as the unique positive number such that Dx(e^x) = e^x
I can answer questions from the standard four semester Calculus sequence. I am not prepared for questions on Tensor Calculus. Everything else is welcome. Derivatives, partial derivatives, ordinary differential equations, single and multiple integrals, change of variable, vector integration (Green`s Theorem, Stokes, and Gauss) and applications.
Ph.D. in Mathematics and many years teaching Calculus at state universities.