To compute the derivative of the function
*f*(*x*)=*e*^{x}, 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 *e*^{x}:

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

The remaining limit (not including the *e*^{x} factor) is
the critical limit for finding the derivative of
*f*(*x*)=*e*^{x} -- if this limit exists and is equal
to *K*, then *f* '(*x*)=*K e*^{x}.
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 e*^{x}
=1 *e*^{x}=*e*^{x}
This formula is important enough to bear repeating:

if *f*(*x*)=*e*^{x}
then *f* '(*x*)=*e*^{x}

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