### Exploration: Exponential Functions and Their Derivatives

Mathlets Main Page
||
Exploration Contents

Back
||
Next

#### Table of Contents

**This exploration uses a small sample of the applets in the
Mathlets package together to explore
exponential functions and their derivatives in depth.**

**Part 1** of this exploration
uses the Exponential Functions applet
to show graphs of the general exponential equation
*y*=*Ce*^{kx}, using the positions of two points
on the graph to determine the values of the constants *C* and *k*.

**Part 2** uses the
Limits
applet, applied to the limit of a difference quotient for the function
*f*(*x*)=*e*^{x}, to compute the derivative
of this exponential function, *f* '(*x*)=*e*^{x}.

**Part 3** uses the
Tangent Lines applet to show how the
derivative formula from Part 2 can be used
to compute the slopes of tangent lines for the exponential function.

**Part 4** uses the
Tangent Lines applet again to
show that the derivative of an exponential function is *not* computed
using the Power Rule for derivatives, by showing that the resulting
slopes do not correspond to tangent lines.

**Part 5** uses the
Differential Equations applet, applied
to the differential equation *y*'=*y* (suggested by the
derivative formula from Part 2) to
show from the graphs
that solutions of this differential equation are, in fact,
exponential functions.
This exploration, and the applets in the
Mathlets package, were written by:

Tom Leathrum

Assistant Professor of Mathematics

Department of Mathematics, Computing, and Information Sciences

Jacksonville State University

Jacksonville, AL

[For copyright information, see the
Mathlets main page.]