DERIVATIVES OF EXPONENTIAL FUNCTIONS
For any fixed postive real number a, there is the exponential function with base a given by y = a^{x}. 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=e^{ln(a)}, we express a^{x} as follows.
a^{x} = (e^{lna})^{x} = e^{xln(a)}
Since ln(a) is a constant, we can use this expression with the chain rule to differentiate the exponential function with base a.
D_{x}(a^{x}) = D_{x}(e^{xln(a)}) = e^{xln(a)}[D_{x}(xln(a))] = a^{x}ln(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=3^{x}.
D_{x}(3^{x}) = 3^{x}ln(3)
Example. Differentiate y=2^{sin(x)}.
D_{x}(2^{sin(x)}) = (2^{sin(x)}ln(2))cos(x).
The final topic of the Stage 6 Lesson is to determine the derivatives of the "other" logarithmic functions.
/Stage6/Lesson/expDeriv.html
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William A. Bogley
Robby Robson