The formula for the derivative of the natural exponential function *f*(*x*)=*e*^{x} was worked out on Page 2 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, on Page 1 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 on Page 1 of this exploration. As examples, look at *y* '=2*y* (*k*=2) and *y* '=*y*/2 (*k*=1/2) in the above applet.