5.4 Exponential Functions: Differentiation and Integration

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5.4 Differentiation and Integration of E 2012
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The Natural Exponential Function

• The function f(x) ln x is increasing on its
  entire domain, and therefore it has an inverse
  function f 1.
• The domain of f 1 is the set of all reals, and
  the range is the set of positive reals, as shown
  in Figure 5.19.

Figure 5.19
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The Natural Exponential Function

• So, for any real number x,
• If x happens to be rational, then
• Because the natural logarithmic function is
  one-to-one, you can conclude that f 1(x) and ex
  agree for rational values of x.

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The Natural Exponential Function

• The following definition extends the meaning of
  ex to include all real values of x.
• The inverse relationship between the natural
  logarithmic function and the natural exponential
  function can be summarized as follows.

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5.4 Exponential FunctionsDifferentiation and
Integration
Solve for x
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5.4 Exponential FunctionsDifferentiation and
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5.4 Exponential FunctionsDifferentiation and
Integration
Definition of Log!
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5.4 Exponential FunctionsDifferentiation and
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5.4 Exponential FunctionsDifferentiation and
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5.4 Exponential FunctionsDifferentiation and
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Chain Rule
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5.4 Exponential FunctionsDifferentiation and
Integration
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5.4 Exponential FunctionsDifferentiation and
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5.4 Exponential FunctionsDifferentiation and
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5.4 Exponential FunctionsDifferentiation and
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Always Positive
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Substitution
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5.4 Exponential FunctionsDifferentiation and
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