Euler's Magic Number (and Differentiation of Exponentials)

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Definition of the derivative
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:

Then, we're interested in the case where . 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 . Let's play with this until we get an explicit definition for e. (The definition we have now is implicit.)

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

The limit is a linear operator, so:

This is one way to define e. If we let , then as . If we apply this, we get a perhaps more well known, but equivalent, definition for e:

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

Exercises:

    Find the derivatives of the following functions.

  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
Last modified 23 June 1997