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Exploration: Exponential Functions and Derivatives - A Simple Differential Equation

Tom Leathrum

The formula for the derivative of the natural exponential function f(x)=ex was worked out on Page 2  of this exploration, and the formula says that f '(x)=ex. This is a curious relationship, though -- f(x) and f '(x) are both the same function, f(x)=f '(x)=ex. This relationship can be expressed as a differential equation by noticing that if y=f(x)=ex, then y '=f '(x)=ex =f(x)=y, or in a more concise form, y '=y. For this differential equation, y '=y, the function y=f(x)=ex is a particular solution, but there are other solutions. For example, if y=g(x)=3ex, then y '=g '(x)=3ex=y also. In fact, the general solution to the differential equation y '=y has the form y=Cex, 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=Cekx also included the undetermined constant k, and a second point was needed to determine both C and k. For a function h(x)=ekx, the derivative h '(x) can be computed using the above formula for the derivative of f(x)=ex, along with the Chain Rule for derivatives, to get h '(x)=kekx, which satisfies the differential equation y '=ky. The general solution of the differential equation y '=ky is y=Cekx, the same equation as on Page 1 of this exploration. As examples, look at y '=2y (k=2) and y '=y/2 (k=1/2) in the above applet.

Tom Leathrum, "Exploration: Exponential Functions and Derivatives - A Simple Differential Equation," Convergence (October 2004)