The following essay considers why the number e, the base of natural
logarithms was discovered.
I made this hypothesis before I learned from John Conway that Napier
computed his first table of natural logarithmic sines in about 1609, about
60 years before the discovery of the calculus.
Anyway, my original guess might still be a possible way for the number e
to be discovered by someone somewhere sometime.
Suppose you wished to find the derivative of the function
f(x) = b^x, where b is a real constant > 1.
We are to evaluate the following expression.
f(x+h) - f(x)
lim ----------------
h--> 0 h
f(x+h) - f(x) = b^(x+h) - b^x
= ( b^x ) ( b^h ) - b^x
= (b^x) ( b^h - 1 )
[ f(x+h) - f(x) ] / h = b^x ( [ b^h - 1 ]/ h )
define g(b,h) = ( b^h - 1) / h
Then ( f(x+h) - f(x) ) / h = f(x) * g(h)
and
f(x+h) - f(x)
f'(x) = lim ----------------
h--> 0 h
= f(x) * lim g(b,h)
h --> 0
Now lets define the function q(b) = lim g(b,h)
h--> 0
This is merely a notational convenience.
f(x) = b^x
f'(x) = b^x * q(b).
The question now is: How do we find the value of q(b) for given b?
b^h - 1
q(b) = lim ----------------
h--> 0 h
Note that
(b1 * b2 )^h - 1
q(b1 * b2)) = lim ----------------
h--> 0 h
Take a closer look at the expression [ (b1 * b2) ^ h - 1 ]/h
[ (b1 * b2)^h - 1] / h
= [ b1^h * b2^h - 1 ] / h
= [ b1^h * b2^h - b2^h + b2^h - 1] / h
= b2^h * [(b1^h - 1)/h ] + [(b2^h - 1) / h)
= b2^h * g(b1,h) + g(b2,h)
q(b1*b2) = q(b1) * lim b2^h + q(b2)
h--> 0
= q(b1) + q(b2)
So the function q is behaving like a logarithm.
f(x) = b^x
f'(x) = b^x * q(b).
If b is such that q(b) = 1, then
f'(x) = f(x).
The number e was defined to be the number such that q(e) = 1.
f(x) = e^x is its own derivative.