Derivative of e^x from first principles

**sakodo** · June 21st 2011, 04:34 AM

Originally Posted by HallsofIvy

That is important for the following reason: If X is any positive real number, then $ln(2^{2X})= 2X ln(2)\ge (2X)(1/2)= X$ . That is, ln(x) is not bounded above.

Could you please explain this line? How does the inequality imply that $ln(x)$ is not bounded above?

Anyway, thanks very much for your detailed response.

The fact that there are so many equivalent definitions kind of bothers me. Its like everything becomes kind of circular. For example when I was learning Analysis(I am still learning it, in fact), one book would define a closed set as the complement of an open set and it would ask me to prove that a closed set contains all of its limit points. Another book would instead define a closed set as a set which contains all of its limit points and ask me to prove that its complement is open. I was like, "What's the point to all this?". I had fun proving it, but it seemed pointless.

**chisigma** · June 21st 2011, 05:09 AM

Originally Posted by sakodo

Thanks for replying.

Looking at posts #3 and #4 by Prove it and chisigma respectively, which one came first? The definition of $e=\lim_{n\to 0} (1+\frac{1}{n})^{n}$ or the defintion of $e^x$ as a function whose derivative is itself? Since if you use either one definition, you can always prove the other.

I think this is what caused my confusion.

In the XVIII° century Leonhard Euler extablished the following definitions for the exponential and logarithm functions of a real variable x ...

$e^{x} = \lim_{n \rightarrow \infty} (1+\frac{x}{n})^{n}$ (1)

... and defined the function $\ln x$ as the inverse of the exponential. The alternative to define...

$\ln x = \int_{1}^{x} \frac{dt}{t}$ (2)

... and the exponential the inverse of logarithm has been proposed in the succesive century so that (1) did come first. Apparently each definition approach is reasonably and there are no reason to prefer one of them... but that is not true if we extend the functions to a complex variable z. Ever Euler discovered the famous formula with his name...

$e^{z}= e^{x + i y} = e^{x} (\cos y + i \sin y)$ (3)

... which permits to define the exponential as a single value function for any value of z. Define the logarithm in the complex field as the inverse of exponential is of course possible but if we do that the function $\ln z$ defined from (3) is a multivalued function , i.e. is defined unless a constant $2 \pi i$ and that is true also if we use the definition...

$\ln z = \int_{1}^{z} \frac{d s}{s}$ (4)

... because the (4) depends from the path connecting the points $s=1$ and $s=z$ ...

The conclusion, in my opinion, is the following: taking the Euler's way You never will have surprises... taking other ways I don't know

...

Kind regards

$\chi$ $\sigma$

**Plato** · June 21st 2011, 06:08 AM

Originally Posted by sakodo

How does the inequality imply that $\ln(x)$ is not bounded above?

Here is the idea. For any $a>0$ we want to show that for some $b>0$ we have $a<\ln(b)$ .
That is, there is no upper bound for $\ln(x)$ .

We know that $\ln(3)>1$ and that implies $a<a\ln(3)=\ln(3^a)$ .
There we have it: $b=3^a$ .

**Deveno** · June 21st 2011, 07:24 AM

Originally Posted by chisigma

In the XVIII° century Leonhard Euler extablished the following definitions for the exponential and logarithm functions of a real variable x ...

$e^{x} = \lim_{n \rightarrow \infty} (1+\frac{x}{n})^{n}$ (1)

... and defined the function $\ln x$ as the inverse of the exponential. The alternative to define...

$\ln x = \int_{1}^{x} \frac{dt}{t}$ (2)

... and the exponential the inverse of logarithm has been proposed in the succesive century so that (1) did come first. Apparently each definition approach is reasonably and there are no reason to prefer one of them... but that is not true if we extend the functions to a complex variable z. Ever Euler discovered the famous formula with his name...

$e^{z}= e^{x + i y} = e^{x} (\cos y + i \sin y)$ (3)

... which permits to define the exponential as a single value function for any value of z. Define the logarithm in the complex field as the inverse of exponential is of course possible but if we do that the function $\ln z$ defined from (3) is a multivalued function , i.e. is defined unless a constant $2 \pi i$ and that is true also if we use the definition...

$\ln z = \int_{1}^{z} \frac{d s}{s}$ (4)

... because the (4) depends from the path connecting the points $s=1$ and $s=z$ ...

The conclusion, in my opinion, is the following: taking the Euler's way You never will have surprises... taking other ways I don't know

...

Kind regards

$\chi$ $\sigma$

this is why defining the exponential function as a power series seems to me, in the long run, the most sensible. it also makes Euler's identity (definition of $e^{iy}$ ) seem less arbitrary. from such a definition, one can quickly recover the differential equations $e^x$ sine and cosine satisfy, as well the values these functions and their derivatives have at 0. moreover, this approach yields an analytic definition of the real numbers $\pi$ and $e$ as certain well-defined rational cauchy sequences, even giving a practical way of calculating (approximations to) them. this in turn, justifies why the function 1/x is a derivative of $exp^{-1}$ . although the integral definition of ln is perfectly adequate for the real numbers, it is hard to understand "why" we would do such a thing (other than it works!).

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