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Exploration: Exponential Functions and Derivatives - Calculating the Derivative

Author(s): 
Tom Leathrum

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 purposes 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

Tom Leathrum, "Exploration: Exponential Functions and Derivatives - Calculating the Derivative," Loci (October 2004)

JOMA

Journal of Online Mathematics and its Applications

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