Calculus 3.
3 Differentiating Inverse Functions Notes
Write your questions
and thoughts here!
Recall:
A function’s inverse is found by swapping the input (𝑥) and output (𝑦) values!
Reciprocal Inverse
Confusing
𝟏 or
Notation: 𝒙 𝟏= 𝒇 𝟏 (𝒙) means inverse of 𝒇.
𝒙
Three ways to say the same thing:
1. 𝑔(𝑥) is the inverse of 𝑓(𝑥)
2. 𝑔(𝑥) = 𝑓 (𝑥)
3. 𝑓 𝑔(𝑥) = 𝑥 and 𝑔 𝑓(𝑥) = 𝑥
Derivative of an Inverse Function:
𝒅 𝟏
[𝒇 𝟏 (𝒙)] =
𝒅𝒙 𝒇 [𝒇 𝟏 (𝒙)]
The table below gives values of the differentiable functions 𝑓, 𝑔, and 𝑓 at selected values of
𝑥. Let 𝑔(𝑥) = 𝑓 (𝑥).
𝒙 𝒇(𝒙) 𝒇 (𝒙)
1 3 −3
2 1 −2
3 −5 −5
4 0 −6
1. What is the value of 𝑔 (1)? 2. Write an equation for the line tangent to
𝑓 at 𝑥 = 1.
3. Let 𝑔 be a differentiable function such that 4. If 𝑓(𝑥) = 3𝑥 + 1 and 𝑔 is the
𝑔(12) = 4, 𝑔(3) = 6, 𝑔 (12) = −5, and inverse function of 𝑓, what is
𝑔 (3) = −2. The function ℎ is differentiable and the value of 𝑔 (25)?
ℎ(𝑥) = 𝑔 (𝑥) for all 𝑥. What is the value of
ℎ (6)?
3.3 Differentiating Inverse Functions
Calculus
Practice
𝟏 (𝒙)
For each problem, let 𝒇 and 𝒈 be differentiable functions where 𝒈(𝒙) = 𝒇 for all 𝒙.
1. 𝑓(3) = −2, 𝑓(−2) = 4, 2. 𝑓(1) = 5, 𝑓(2) = 4,
𝑓 (3) = 5, and 𝑓 (−2) = 1. 𝑓 (1) = −2, and 𝑓 (2) = −4.
Find 𝑔 (−2). Find 𝑔 (5).
3. 𝑓(6) = −2, 𝑓(−3) = 7, 4. 𝑓(−1) = 4, 𝑓(2) = −3,
𝑓 (6) = −1, and 𝑓 (−3) = 3. 𝑓 (−1) = −5, and 𝑓 (2) = 7.
Find 𝑔 (7). Find 𝑔 (−3).
The table below gives values of the differentiable function 𝒈 and its derivative 𝒈 at selected values of 𝒙.
Let 𝒉(𝒙) = 𝒈 𝟏 (𝒙).
𝒙 𝒈(𝒙) 𝒈 (𝒙)
−1 −2 −4
−2 −5 −2
−3 −4 −1
−4 −3 −5
−5 −1 −3
5. Find ℎ (−1) 6. ℎ (−3) 7. ℎ (−5)
Find the equation of the tangent Find the equation of the tangent Find the equation of the tangent
line to 𝑔 at 𝑥 = −1. line to 𝑔 at 𝑥 = −3. line to 𝑔 at 𝑥 = −5.
𝒇 and 𝒈 are differentiable functions. Use the table to answer the problems below. 𝒇 and 𝒈 are
NOT inverses!
𝑥 𝑓(𝑥) 𝑓 (𝑥) 𝑔(𝑥) 𝑔 (𝑥)
1 5 −5 4 5
2 1 −6 3 3
3 6 4 1 6
4 2 9 6 1
5 3 1 1 2
6 4 2 2 4
8. 𝑔 (4) 9. 𝑓 (5) 10. 𝑔 (3)
11. 𝑓 (1) 12. Find the line tangent to the graph of 𝑓 (𝑥) at
𝑥 = 2.
𝟏 (𝒙)
For each function 𝒈(𝒙), its inverse 𝒈 = 𝒇(𝒙). Evaluate the given derivative.
13. 𝑔(𝑥) = cos(𝑥) + 3𝑥 14. 𝑔(𝑥) = 2𝑥 − 𝑥 − 5𝑥
𝑔 = . Find 𝑓 𝑔(−2) = −10. Find 𝑓 (−10)
15. 𝑔(𝑥) = √8 − 2𝑥. Find 𝑓 (4)? 16. 𝑔(𝑥) = 𝑥 − 7. Find 𝑓 (20)? 17. 𝑔(𝑥) = . Find 𝑓 ?
3.3 Differentiating Inverse Functions Test Prep
18. The functions 𝑓 and 𝑔 are differentiable for all real numbers and 𝑔 is strictly increasing. The table
below gives values of the functions and their first derivatives at selected values of 𝑥. The function ℎ
is given by ℎ(𝑥) = 𝑓 𝑔(𝑥) − 6.
𝑥 𝑓(𝑥) 𝑓 (𝑥) 𝑔(𝑥) 𝑔 (𝑥)
1 6 4 2 5
2 9 2 3 1
3 10 −4 4 2
4 −1 3 6 7
(a) Explain why there must be a value 𝑟 for 1 < 𝑟 < 3 such that ℎ(𝑟) = −5.
(b) If 𝑔 is the inverse function of 𝑔, write an equation for the line tangent to the graph of
𝑦 = 𝑔 (𝑥) at 𝑥 = 2.
19. A function ℎ satisfies ℎ(3) = 5 and ℎ (3) = 7. Which of the following statements about the inverse
of ℎ must be true?
(A) (ℎ ) (5) = 3 (B) (ℎ ) (7) = 3 (C) (ℎ ) (5) = 7
(D) (ℎ ) (5) = (E) (ℎ ) (7) =
