We know that aⁿ exists for all n which are positive or negative integers. aⁿ=a*a*a...*a (n times) and a^-n=1/aⁿ. (As a special case, a⁰=1, if a does not vanish.)

If n is rational (n=p/q), aⁿ = (a^p)^1/q

If r is irrational, we look at a sequence of rational numbers x->r, and define

a^r = lim_x->ra^x, x rational

If a > 1, then a^x is an increasing function of x.

If a < 1, then a^x is a decreasing function of x.

If a=1, then a^x is a constant function of x.

Note: a^x, a > 1, grows faster than any function x^p for any value of p!

Computing the derivative of f(x) = a^x :

At this point, we must use the limit definition of the derivative

f'(x) = lim_h->0 [f(x+h)-f(x)]/h

= lim_h->0 [a ^x+h - a^x]/h

= lim_h->0 a^x[a^h-1]/h

= a^x lim_h->0 [a^h-1]/h

The term multiplying a^x is a constant, and is equal to f'(0) in fact, so we have

f'(x) = f'(0) a^x = f'(0)*f(x)

that is, the rate of change of the function is proportional to the function itself!

This fact uniquely characterizes the exponential function, no other continuous function has this property!

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For some value of a, f'(0)=lim_h->0 [a^h-1]/h = 1, we call this value e=2.71828182845904523536... It is irrational (nonrepeating), and transcendental (solution of no algebraic equation).

In this case, a=e, d/dx e^x = e^x. Applying the chain rule,

d/dx e ^u = e^u du/dx

We have the following important limits, lim_x->infinity e^x=infinity, and lim_x->-infinity e^x=0.

Last modified Wed Oct 16 22:23:56 CDT 1996