Differentiable Exponential Functions

The exponential property is actually a rather stringent condition on a differentiable function $f:\left(0,\infty\right)\rightarrow\mathbb{R}$ . Let’s start by assuming that $f$ is a differentiable exponential function and see what happens.

We calculate the derivative as usual by taking the limit of the difference quotient

$\displaystyle f'(x)=\lim\limits_{\Delta x\rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

Then the exponential property says that our derivative is

$\displaystyle f'(x)=\lim\limits_{\Delta x\rightarrow0}\frac{f(x)f(\Delta x)-f(0)f(x)}{\Delta x}=f(x)\lim\limits_{\Delta x\rightarrow0}\frac{f(\Delta x)-f(0)}{\Delta x}=f(x)f'(0)$

So we have a tight relationship between the function and its own derivative. Let’s see what happens for the exponential function $\exp$ . Since it’s the functional inverse of $\ln$ we can use the chain rule to calculate

$\displaystyle\exp'(y)=\frac{1}{\ln'(\exp(y))}=\frac{1}{\frac{1}{\exp(y)}}=\exp(y)$

Showing that this function is its own derivative. That is, this is the exponential function with $f'(0)=1$ .

Since a general (differentiable) exponential function $f$ is a homomorphism from the additive group of reals to the multiplicative group of positive reals, we can follow it by the natural logarithm. This gives a differentiable homomorphism from the additive reals to themselves, which must be multiplication by some constant $C_f$ . That is: $\ln(f(x))=C_fx$ . How can we calculate this constant? Take derivatives!

$\displaystyle\frac{d}{dx}\ln(f(x))=\frac{1}{f(x)}f'(x)=\frac{f(x)f'(0)}{f(x)}=f'(0)$

So our constant is the derivative $f'(0)$ from before. Of course we could also write

$\ln(f(x))=f'(0)x=\ln(\exp(f'(0)x))$

And since $\ln$ is invertible this tells us that $f(x)=\exp(f'(0)x)$ . That is, every differentiable exponential function comes from $\exp$ by taking some constant multiple of the input.

By the usual yoga of inverse functions we can then see that every differentiable logarithmic function (an inverse to some differentiable exponential function) is a constant multiple of the natural logarithm $\ln$ . That is, if $g(x)$ satisfies the logarithmic property, then $g(x)=C_g\ln(x)$

April 10, 2008 - Posted by John Armstrong | Analysis, Calculus

7 Comments »

I have been thinking of starting a series of posts titled “Category Theory for Dummies”! And, I desperately need to know if there is any open-source software that I can use to create diagrams – plenty of ’em. I think you mentioned once earlier that you use an old version of Maple or something for that purpose. But, what about other software? I have searched quite a bit on the Internet, but couldn’t find an appropriate one for the purpose I just mentioned above.

(Sorry for asking this question this over here!)

Comment by Vishal Lama | April 11, 2008 | Reply
Ah, never mind! I think I may have finally found one: Inkscape.

Comment by Vishal Lama | April 11, 2008 | Reply
Arrgh! Inkscape won’t let me write subscripts (such as in $1_A$ ) though otherwise it is a wonderful piece of software!

Comment by Vishal Lama | April 11, 2008 | Reply
The diagrams I posted were made with the “commutative diagrams” package for LaTeX in TeXShop, then screencaptured and cleaned up a bit.

Comment by John Armstrong | April 12, 2008 | Reply
Thanks a lot for that tip! I should have thought about it earlier myself. I found out that one can use the XYPic package to draw all sorts of diagrams, especially the ones we use when working with categories. My diagrams aren’t as neat as yours, but they are decent enough for my posts.

Comment by Vishal Lama | April 12, 2008 | Reply
[…] one that’s particularly nice is the exponential function . We know that this function is its own derivative, and so it has infinitely many derivatives. In particular, , , , …, , and so […]

Pingback by The Taylor Series of the Exponential Function « The Unapologetic Mathematician | October 7, 2008 | Reply
[…] value gets, the faster it grows. That is, the exponential function satisfies the equation . We already knew this about , but there we ultimately had to use the fact that we defined the logarithm to have a […]

Pingback by The Exponential Differential Equation « The Unapologetic Mathematician | October 10, 2008 | Reply

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