Derivatives of Inverse Functions AP Calculus Unit 6 Lesson 3 Mrs. Mongold

Continuity and Differentiability of Inverse Functions

The Derivative of An Inverse Function Let f be a function that is differentiable on an interval I . If f has an inverse function g, then g is differentiable at any x for which f’(g(x)) does not equal 0. Moreover, 𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 )

Example Let f(x) = ¼ x3 + x – 1 What is the value of f-1(x) when x = 3? What is the value of (f-1)’(x) when x = 3?

Example Solution We know an inverse exists. Use calculator to find out when f(x) = 3.

Example Solution We know an inverse exists. Use calculator to find out when f(x) = 3. We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2

Example Solution We know an inverse exists. Use calculator to find out when f(x) = 3. We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2 Because f is differentiable and has an inverse you can use 𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 )

Example Solution We know an inverse exists. Use calculator to find out when f(x) = 3. We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2 Because f is differentiable and has an inverse you can use 𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 ) 𝑓 −1 ′ 3 = 1 𝑓 ′ ( 𝑓 −1 3 ) = 1 𝑓 ′ (2)

Example Solution We know an inverse exists. Use calculator to find out when f(x) = 3. We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2 Because f is differentiable and has an inverse you can use 𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 ) 𝑓 −1 ′ 3 = 1 𝑓 ′ ( 𝑓 −1 3 ) = 1 𝑓 ′ (2) = 1 3 4 2 2 +1

Example Solution We know an inverse exists. Use calculator to find out when f(x) = 3. We know that f(x) = 3 when x = 2 so we know that f-1(3) = 2 Because f is differentiable and has an inverse you can use 𝑔 ′ 𝑥 = 1 𝑓′(𝑔 𝑥 ) 𝑓 −1 ′ 3 = 1 𝑓 ′ ( 𝑓 −1 3 ) = 1 𝑓 ′ (2) = 1 3 4 2 2 +1 = 1 4

Homework Verify f has an inverse then use the function and the given real number to find (f-1)(a) 1. f(x) = x3 – 1, a=26 2. f(x) = x3 + 2x – 1, a = 2 3. f(x) = sinx, −𝜋 2 ≤𝑥≤ 𝜋 2 , a = ½ 4. f(x) = 𝑥+6 𝑥−2 , x>2, a = 3 5. f(x) = 𝑥−4 , 𝑎=2