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- Thread starter Cisco Qid
- Start date

(ax)' = f'(x) = Lim (h -> 0) (ax+h - ax)/ h = ax Lim h->0 (ah - 1)/h

In this case, we are interested values of a where: Lim h->0 (ah - 1)/h = 1

or (ah - 1)/h = 1

Solve for a: ah - 1 = h and ah = 1 + h

or a = (1+h)1/h

or a = Lim h->0 (1+h)1/h

Let n = 1/h then n goes to infinity as h goes to zero or:

a = Lim n-> infinity (1 + 1/n)n This is the definition of Euler's constant, e.

Thus if a is equal to Euler's constant e then f(x) = f'(x) = ex where

e = Lim n-> infinity (1 + 1/n)n

I developed this definition for e, some years back

I know that I was frustrated when they simply gave the answer f'(x) = exp(x) where e = limit as n goes to infinity of (1+1/n)^n. No matter which Calculus book, my question was always, why?

Why?

Here is an explanation using a

I'm comfortable with a factorial definition as a series. "It's the

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My math editor doesn't work here.

On the subject of the

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