# 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

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?