Derivatives of Inverse Functions Lesson 3 6 Terminology

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 … 1. If f(x) is continuous on its domain, then f -1(x) is continuous on its domain

Continuity and Differentiability Furthermore … 2. If f(x) is differentiable at c and f '(c) ≠ 0 then f -1(x) is differentiable at f(c) f(x) Note the counter example • f(x) not differentiable here f -1(x) • 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 reciprocals they are

Derivatives of Inverse Trig Functions Note further patterns on page 177

Practice • Find the derivative of the following functions

More Practice • Given • Find the equation of the line tangent to this function at

Assignment • Lesson 3. 6 • Page 179 • Exercises 1 – 49 EOO, 67, 69