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Derivatives of Inverse Functions. Lesson 3.6. 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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terminology
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
continuity and differentiability
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
continuity and differentiability1

f(x)

f -1(x)

Continuity and Differentiability

Furthermore …

  • If f(x) is differentiable at cand f \'(c) ≠ 0then f -1(x) is differentiable at f(c)
  • Note the counter example
  • f(x) not differentiable here
  • f -1(x) not differentiable here
derivative of an inverse function
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

we gotta try this
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)

consider this phenomenon
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
derivatives of inverse trig functions
Derivatives of Inverse Trig Functions

Note further patterns on page 177

practice
Practice
  • Find the derivative of the following functions
more practice
More Practice
  • Given
  • Find the equationof the line tangentto this function at
assignment
Assignment
  • Lesson 3.6
  • Page 179
  • Exercises 1 – 49 EOO, 67, 69
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