Euler's Magic Number (and Differentiation of Exponentials)

Suggested Prerequesties:

Definition of the derivative, Properties of exponentials and logarithms, Evaluating limits

One rainy day, Leonhard Euler, everyone's favorite mathematician, got bored and decided that it would be really cool to find a function that was its own derivative. (OK maybe that's not really how it happened, but I like to think of it that way =) So, on this wonderful quest for knowledge that everyone seemed to be doing in the eighteenth century, he decided that an exponential function (f(x) = ax for some base a) was the most likely candidate for this special attribute. In order to find this special function, he turned to the definition of the derivative:
ddxf(x) &sp;=&sp; lim h → 0 f(x  +  h)  -  f(x) h

ddxa^x^ &sp;=&sp; lim h → 0 a^x+h^  -  a^x^ h &sp;=&sp; lim h → 0 a^x^a^h^  -  a^x^ h

&sp;=&sp; lim h → 0 ( a^x^   a^h^  -  1 h ) &sp;=&sp; a^x^   ( lim h → 0 a^h^  -  1 h )

Then, we're interested in the case where lim h → 0 e^h^  -  1 h &sp;=&sp; 1. Let's call the value of a for which this is true e, in honor of Euler. (There's a really cool discussion of this by Douglas Arnold here. Take a look at it.)

So, now we want to find the value of e that satisfies lim h → 0 e^h^  -  1 h &sp;=&sp; 1. Let's play with this until we get an explicit definition for e. (The definition we have now is implicit.)

lim h → 0 ( e^h^  -  1 h ) &sp;=&sp; 1

lim h → 0 ( e^h^  -  1 h ) &sp;=&sp; lim h → 0 1

Since the limit of a product equals the product of the limit (and vice versa) we can make the following transformation:

lim h → 0 (e^h^  -  1 ) &sp;=&sp; lim h → 0 h

The limit is a linear opperator, so:

lim h → 0 e^h^ &sp;=&sp; lim h → 0 ( 1  +  h )

e &sp;=&sp; lim h → 0 ( 1  +  h ) ^ 1h ^

This is one way to define e. If we let n  =  1h , then n → &infty; as h → 0. If we apply this, we get a perhaps more well known, but equivalent, definition for e:

e &sp;=&sp; lim n → &infty; ( 1  +  1n ) ^n^

In words, this says that e is the limit of 1 plus a really small number, raised to a really big number. We can use this to make rough approximations of e :
for n = 1 e = 2
for n = 10 e = 2.5937425
for n = 100 e = 2.7048138
for n = 1000 e = 2.7169239
People who either had lots of time on their hands, or good computers, or both, have produced this accepted approximation for e: e = 2.7 1828 1828 45 90 45 . Impress your friends by knowing e to 15 decimal places, by thinking of it in these easy to remember groupings.

So, now we have a useful definition of e, Euler's magic number. We also know that Dxex = ex, since that was our starting point. Let's works some examples using this information:
  1. Dxex = ex (duh)

  2. Remember the chain rule?

    Dxe3x = 3 ex

  3. Dx(ex)2 = (2 ex) (ex) = 2 e2x

    or alternitavely Dx(ex)2 = Dxe2x = 2 e2x

  4. D_x_ e^sin x^ &sp;=&sp; (cos x)e^sin x^

Exercises:

    Find the derivatives of the following functions.

  1. y &sp;=&sp; 12 ( e^x^  +  e^-x^ )

  2. f(x) = x2ex2

  3. h(x) = xe + ex + e

Solutions to the exercises | Back to the Calculus page | Back to the World Web Math top page

jjnichol@mit.edu