### Exploration: Exponential Functions and Their Derivatives

#### Part 2: Calculating the Derivative

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To compute the derivative of the function f(x)=ex, use the definition of the derivative as a limit of a difference quotient:

This limit can be simplified using properties of exponents:

Now, in the numerator, factor out the common ex:

Since the limit is with respect to the variable h, and the variable x is independent of h, the ex may be treated as a constant factor and removed completely from the limit:

The remaining limit (not including the ex factor) is the critical limit for finding the derivative of f(x)=ex -- if this limit exists and is equal to K, then f '(x)=K ex. It remains only to find K (if it exists). For the purpose of plugging into the applet below to find K, rewrite the limit in terms of the variable x instead of h:

Now use the tables in the applet below to approximate the value of K.

How to use this applet
From these tables, it seems that K=1. This is an approximation to K using the values in the tables from the applet, but it seems to be a good approximation, given the trends in the tables. If this is the case, then (from above)
f '(x)=K ex =1 ex=ex
This formula is important enough to bear repeating:
if f(x)=ex then f '(x)=ex

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