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
Graphs

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
d

dx
au = (auln(a))
du

dx

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.


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William A. Bogley
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