Question about the derivative of e^x

1. Oct 6, 2008

The_ArtofScience

I've been drawn to this expression for a long time.

Anyway, my question is - is the derivative of e^x exact? That is with absolute precision, the derivative of e^x is EXACTLY e^x?

Sorry if this seems like a silly question, I would really like to know if my math instructor is right on this one

I remember the argument:

lim h-->0 e^x+h - e^x/ h

What about the constant e? Is it approximate or absolutely precise?

Say you have a base, b.

Then lim q->0 ([b^1+q -b]/q) = b, which through some adjustments gives e = lim n-> infinity (1 +1/n)^n

Last edited: Oct 7, 2008
2. Oct 7, 2008

CompuChip

Yes. In general, when you derive $g^x$, with g any number, the derivative is
$$\operatorname{ln}(g) g^x = k \log(g) g^x$$, where "ln" is some weird function which turns out to be the logarithm (in base 10), up to a multiplicative constant k.
Now when you plug in some numbers for g, you will find that this function "ln" comes close to unity if you take g to be somewhere around 2.7. Now it would be very cool to know this number g for which g^x is its own derivative exactly, but it turns out not to be anything nice (for example, it is not a fraction, or a square root of something). So we define e to be this number, for which
$$\operatorname{ln}(e) = 1$$
and then of course (by the properties of the logarithm) the constant k is
$$k = 1/\operatorname{{}^{10}log}(e)$$.

So, by (one the many equivalent) definition of e, the derivative of e^x is e^x. As long as you write e and not some numerical approximation like 2.71828... this is an exact identity. Indeed, by definition of the derivative one can show that this requirement of e^x being its own derivative is equivalent to using
$$e \stackrel{\text{def}}{\equiv} \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n$$.

3. Oct 7, 2008

HallsofIvy

Staff Emeritus
Yes, the number e, like any number, is exact. The derivative of the function ex is exactly ex. Pretty nice function, eh?

4. Oct 7, 2008

matematikawan

The coolest function ever defined in calculus is ex. You differentiate or integrate this function you get back the function. Who actually the first to defined such uninteresting function?