### DERIVATIVE OF THE EXPONENTIAL FUNCTION

It is a very natural question to ask if there is a function f that is equal to its own derivative.

f '(x) = f(x)

 A review of exponential functions is available in the Field Guide to Functions.
It turns out that there are such functions. Indeed, any constant multiple of the exponential function is equal to its own derivative.

f(x) = Cex

Here C is any fixed real constant and e is Euler's irrational number.

e 2.7182818...

The number e also occurs as a certain limit. You investigated the following limit statement in Stage 3.

 lim [1 + t]1/t = e t->0

The exponential function is sometimes denoted this way.

exp(x) = ex

Here are some summary facts about the exponential function.

• The exponential function is differentiable on the entire real line.
• ex > 0 for all real numbers x.
• ex+h = exeh for all real numbers x and h
• e0 = 1
• (ex)c = exc for all real numbers x and c.
Here are three limit statements concerning the exponential function. The first says that ex can be made arbitrarily large by choosing x to be sufficiently large.

 lim ex = + x-> +

The second statement says that the graph of the exponential function has a horizontal asymptote at y=0 (the x-axis).

 lim ex = 0 x-> -

A graphical interpretation of the third limit statement may not be immediately obvious. We will use this limit statement to compute the derivative of the exponential function. (Hint: Think difference quotient!)
 lim h->0
 eh - 1 h
= 1

The exponential function is equal to its own derivative. In order to verify this claim, we examine the limit of the difference quotient. What else?!! Fix a real number x.

Dx(ex) =
 lim h->0
 ex+h - ex h

We use properties of the exponential function to simplify the difference quotient. In particular, bear in mind that ex+h=exeh and e0=1.

 ex+h - ex h
= ex (
 eh - 1 h
)

Now, ex is constant as h approaches 0, so we complete the calculation as follows.

Dx(ex) =
 lim h->0
 ex+h - ex h
=
 lim h->0
ex (
 eh - 1 h
)
=
ex (
 lim h->0
 eh - 1 h
)
= ex(1) = ex

With our three basic derivative formulas in hand, we now move on to general techniques of differentiation.

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