Derivatives of Inverse Functions

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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 &quot; f-inverse of R“ is not an exponent it does not mean reciprocal .

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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
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

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

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!
• 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
• 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

Note further patterns on page 177

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