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)
It turns out that there are such functions. Indeed, any constant multiple of the exponential function is equal to its own derivative.
f(x) = Ce^{x}
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.
y = exp(x) = e^{x}

The exponential function is sometimes denoted this way.
exp(x) = e^{x}
Here are some summary facts about the exponential function.
 The exponential function is differentiable on the entire real line.
 e^{x} > 0 for all real numbers x.
 e^{x+h} = e^{x}e^{h} for all real numbers x and h
 e^{0} = 1
 (e^{x})^{c} = e^{xc} for all real numbers x and c.
Here are three limit statements concerning the exponential function. The first says that e^{x} can be made arbitrarily large by choosing x to be sufficiently large.
lim  e^{x}  =  + 
x> +   

The second statement says that the graph of the exponential function has a horizontal asymptote at y=0 (the xaxis).
lim  e^{x}  =  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!)
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.
We use properties of the exponential function to simplify the difference quotient. In particular, bear in mind that e^{x+h}=e^{x}e^{h} and e^{0}=1.
Now, e^{x} is constant as h approaches 0, so we complete the calculation as follows.
D_{x}(e^{x}) 
= 


= 


= 


= 
e^{x}(1) = e^{x} 
With our three basic derivative formulas in hand, we now move on to general techniques of differentiation.
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© CalculusQuest1996
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Version 1
William A. Bogley
Robby Robson