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) = bx be the exponential function. Then the derivative of f is f '(x) = bx 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 = ex 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) = ex is f '(x) = ex

• 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.