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# Derivatives of Inverse Functions - PowerPoint PPT Presentation

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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### Derivatives of Inverse Functions

Lesson 3.6

• 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

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

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

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

• Given

• g(2) = 2.055 and

• So

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

• 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

Note further patterns on page 177

• Find the derivative of the following functions

• Given

• Find the equationof the line tangentto this function at

• Lesson 3.6

• Page 179

• Exercises 1 – 49 EOO, 67, 69