Objectives: In this tutorial, the derivative of the general exponential
function is obtained. The formula is written in terms of the derivative at
x = 0. Using this formula, the number e is defined.
After working through these materials, the student should be able to
derive the formula for the derivative of the exponential function.
Modules:
Theorem. Let f(x) = b^{x} be the
exponential function. Then the derivative of f is
f '(x) = b^{x} f '(0)


 Discussion
[Using HotEqn]
[Using IBM Techexplorer]
[Using IBM Pro. Techexplorer]
Definition. e is the real number such that the slope of the
tangent line to the graph of the exponential function y = e^{x} at x = 0 is 1.


 LiveMath notebook which can be used to calculate e using the definition above.
Corollary. The derivative of the exponential function f(x) = e^{x} is
f '(x) = e^{x}


 The derivative of the other exponential functions will be found in a later module.
 A Javascript numerical exploration
of the value of f '(0) when f(x) = a^{x}.