Talk:Derivative of Exponential Function

$$\frac{dy}{dx}=\frac{e^x(e^h-1)}{h}$$

$$\lim_{h \to 0}{(e^h-1)}=h$$

$$\frac{dy}{dx}=\frac{e^xh}{h}$$

'Is the second line above correct? It seems to me that as h → 0, eh→ 1 and (eh --1)→ 0'

Definition of e
According to Stewart's Calculus: Early Transcendentals, the definition of the number e is the number such that its derivative at x=0 is 1.

This is equivalent to stating $$~\lim_{h \to 0} \frac{e^h - 1}{h} = 1 $$.

Why? This is simply because

$$f(x)=e^x\rightarrow f^'(x)=\lim_{h \to 0}\frac{e^{x+h}-e^x}{h} = \lim_{h \to 0}\frac{e^x(e^h - 1)}{h}$$

Defining $$f^'(0)=1$$, we get

$$ 1 = f^'(0) = \lim_{h \to 0}\frac{e^0(e^h - 1)}{h} = \lim_{h \to 0}\frac{1\cdot(e^h-1)}{h}=\lim_{h \to 0}\frac{e^h-1}{h}$$

There ya go.

Infinite Series
Take a common definition of the exponential function: $$e^x = \sum_{n = 0}^{\infty} {\frac{x^n}{n!}}$$

Evaluating the limit based on this definition: $$\lim_{h \to 0} \frac{e^h - 1}{h}=\lim_{h \to 0} \frac{\sum_{n = 0}^{\infty} {\frac{h^n}{n!}}-1}{h}$$

$$=\lim_{h \to 0} \frac{\frac{h^0}{0!}+\sum_{n = 1}^{\infty} {\frac{h^n}{n!}}-1}{h}$$

$$=\lim_{h \to 0} \frac{1+\sum_{n = 1}^{\infty} {\frac{h^n}{n!}}-1}{h}$$

$$=\lim_{h \to 0} \sum_{n = 1}^{\infty} {\frac{h^{n-1}}{n!}}$$

$$=\lim_{h \to 0} \sum_{n = 0}^{\infty} {\frac{h^{n}}{(n+1)!}}$$

$$=\sum_{n = 0}^{\infty} {\frac{0^n}{(n+1)!}}$$

$$=\frac{0^0}{(0+1)!}+\sum_{n = 1}^{\infty} {\frac{0^n}{(n+1)!}}$$

$$=\frac{1}{1}+0$$

$$=1$$

Question
This might be a little too picky and it's not even on the main proof page but "definition of the number e is the number such that its derivative at x=0 is 1" clearly can't be correct. Do you mean the function $$ a^x $$ with base $$e$$?

One assumes, but you've caught me without my copy of Stewart's Calculus: Early Transcendentals, so I can't check. That does seem to be a fair definition of the exponential function, though. See above for a proof based on an alternate definition of $$e^x$$. Note that this talk page is getting kind of long (longer than many of the proof pages[yeah, largely my fault :D]). We might want to put some of it on the page itself. --Cynic 00:50, 25 June 2008 (UTC)