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Derivatives of Inverse Functions

This lesson covers the terminology and concepts related to derivatives of inverse functions, including continuity, differentiability, and finding the derivative of inverse trigonometric functions.

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Derivatives of Inverse Functions

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  1. Derivatives of Inverse Functions Lesson 3.6

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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)

  7. 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

  8. Derivatives of Inverse Trig Functions Note further patterns on page 177

  9. Practice • Find the derivative of the following functions

  10. More Practice • Given • Find the equationof the line tangentto this function at

  11. Assignment • Lesson 3.6 • Page 179 • Exercises 1 – 49 EOO, 67, 69

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