DERIVATIVES OF EXPONENTIAL FUNCTIONS
For any fixed postive real number a, there is the exponential function with base a given by y = ax. The exponential function with base e is THE exponential function. The exponential function with base 1 is the constant function y=1, and so is very uninteresting. The graphs of two other exponential functions are displayed below.
Exponential Functions with Base 2 and 1/2
All of the exponential functions can be expressed in terms of THE exponential function. Let a positive real number a be chosen once and for all. Using the fact the a=eln(a), we express ax as follows.
ax = (elna)x = exln(a)
Since ln(a) is a constant, we can use this expression with the chain rule to differentiate the exponential function with base a.
Dx(ax) = Dx(exln(a)) = exln(a)[Dx(xln(a))] = axln(a)
Combining with the chain rule for future use, here is the general result. Here, a is a fixed positive real number and u is a differentiable function of x.
Derivatives of Exponential Functions
Example. Differentiate y=3x.
Dx(3x) = 3xln(3)
Example. Differentiate y=2sin(x).
Dx(2sin(x)) = (2sin(x)ln(2))cos(x).
The final topic of the Stage 6 Lesson is to determine the derivatives of the "other" logarithmic functions.
All rights reserved---1996
William A. Bogley