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Derivatives of Inverse Functions

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g(x) is differentiable for any x where. f '(g(x)) 0. And ... f '(g(x)) 0. We ... By the definition they are reciprocals. Derivatives of Inverse Trig Functions ... – PowerPoint PPT presentation

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Title: Derivatives of Inverse Functions


1
Derivatives of Inverse Functions
  • Lesson 3.6

2
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

3
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

4
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

5
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
6
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)
7
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

8
Derivatives of Inverse Trig Functions
Note further patterns on page 177
9
Practice
  • Find the derivative of the following functions

10
More Practice
  • Given
  • Find the equationof the line tangentto this
    function at

11
Assignment
  • Lesson 3.6
  • Page 179
  • Exercises 1 49 EOO, 67, 69
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