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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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