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
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