The formula for the derivative of the natural exponential function
*f*(*x*)=*e*^{x} was worked out in
previous parts of this exploration, and
the formula says that *f* '(*x*)=*e*^{x}.
This is a curious relationship, though -- *f*(*x*) and
*f* '(*x*) are both the same function,
*f*(*x*)=*f* '(*x*)=*e*^{x}.
This relationship
can be expressed as a *differential equation* by noticing that
if *y*=*f*(*x*)=*e*^{x},
then *y* '=*f* '(*x*)=*e*^{x}
=*f*(*x*)=*y*, or in a more concise form,
*y* '=*y*. For this differential equation,
*y* '=*y*, the function
*y*=*f*(*x*)=*e*^{x} is a
*particular solution*, but there are other solutions. For example,
if *y*=*g*(*x*)=3*e*^{x}, then
*y* '=*g* '(*x*)=3*e*^{x}=*y*
also. In fact, the *general solution* to the differential
equation *y* '=*y* has the form
*y*=*Ce*^{x}, where *C* is a constant.
The value of *C* can be determined by knowing the position of
*one* point on the graph of the solution function, as can be
seen with the differential equation *y* '=*y* in
the following applet:

How to use this applet

However, in the first part of this exploration,
the graphs of exponential functions required *two* points to
determine the graph -- this was because the equation there,
*y*=*Ce*^{kx} also included the undetermined
constant *k*, and a second point was needed to determine both
*C* and *k*. For a function
*h*(*x*)=*e*^{kx}, the derivative
*h* '(*x*) can be computed using the above formula
for the derivative of *f*(*x*)=*e*^{x},
along with the Chain Rule for derivatives, to get
*h* '(*x*)=*ke*^{kx}, which
satisfies the differential equation *y* '=*ky*.
The general solution of the differential equation *y* '=*ky*
is *y*=*Ce*^{kx}, the same equation as in the
first part of this exploration.
As examples, look at
*y* '=2*y* (*k*=2) and
*y* '=*y*/2 (*k*=1/2)
in the above applet.

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