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Derivatives of Inverse FunctionsPowerPoint Presentation

Derivatives of Inverse Functions

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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 -1(x)

Continuity and DifferentiabilityFurthermore …

- 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

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