Derivatives of Inverse Functions

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Derivatives of Inverse Functions

• Lesson 3.6

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Terminology

• If R f(T) ... resistance is a function of
  temperature,
• Then T f -1(R) ... temperature is the inverse
  function of resistance.
• f -1(R) is read "f-inverse of R
• is not an exponent
• it does not mean reciprocal

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Continuity and Differentiability

• Given f(x) a function
• Domain is an interval I
• If f has an inverse function f -1(x) then
• If f(x) is continuous on its domain, thenf
  -1(x) is continuous on its domain

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Continuity and Differentiability

• Furthermore
• If f(x) is differentiable at c and f '(c) ? 0
  then f -1(x) is differentiable at f(c)

• Note the counter example
• f(x) not differentiable here
• f -1(x) not differentiable here

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Derivative of an Inverse Function

• Given f(x) a function
• Domain is an interval I
• If f(x) has an inverse g(x) then g(x) is
  differentiable for any x where f '(g(x)) ? 0
• And

f '(g(x)) ? 0
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We Gotta Try This!

• Given
• g(2) 2.055 and
• So

Note that we did all this without actually taking
the derivative of f -1(x)
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Consider This Phenomenon

• For(2.055, 2) belongs to f(x)(2, 2.055)
  belongs to g(x)
• What is f '(2.055)?
• How is it related to g'(2)?
• By the definition they
  are reciprocals

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Derivatives of Inverse Trig Functions
Note further patterns on page 177
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Practice

• Find the derivative of the following functions

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More Practice

• Given
• Find the equationof the line tangentto this
  function at

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Assignment

• Lesson 3.6
• Page 179
• Exercises 1 49 EOO, 67, 69