1. 2014 Derivatives of Inverse Functions
AP Calculus

2. Monotonic – always increasing or always decreasing
Inverses
Existence of an Inverse: If f(x) is one-to-one on its domain D , then f is called invertible. Further,
Domain of f = Range of f -1
Range of f = Domain of f -1
𝑦= 𝑥 2
𝑆𝑤𝑎𝑝 𝑥 𝑎𝑛𝑑 𝑦
𝑥= 𝑦 2
± 𝑥 =𝑦
One-to One Functions: A function f(x) is one-to one (on its domain D) if for every x there exists only one y
and for every y there exists only one x
Horizontal line test.

3. Find the inverse 𝑦= 𝑥+3 𝑥+1 𝑥= 𝑦+3 𝑦+1 Switch x and y multiply
𝑥 𝑦+1 =𝑦+3
𝑥𝑦+𝑥=𝑦+3
distribute
Collect y
𝑥𝑦−𝑦=3−𝑥
𝑦 𝑥−1 =3−𝑥
factor
𝑦= 3−𝑥 𝑥−1
divide

4. Find the inverse
𝑦= 3 𝑥+4
𝑥= 3 𝑦+4
𝑥 3 =𝑦+4
𝑥 3 −4=𝑦

5. 𝑦= 𝑥 2 −4
for x ≥ 2 makes it monotonic

6. REVIEW: Inverse Functions
If f(x) is a function and ( x, y) is a point on f(x) , then the inverse f -1(x) contains the point ( y, x)
To find f -1(x)
Reverse the x and y and resolve for y.
(a,b)
(b,a)
Theorem:
f and g are inverses iff
f(g(x)) = g(f(x)) = x

7. 𝑓 𝑥 = 𝑥 3 −4
𝑔 𝑥 = 3 𝑥+4
𝑓 𝑔 𝑥 =𝑔 𝑓 𝑥
3 𝑥 −4= 3 𝑥 3 −4+4
𝑥+4−4= 3 𝑥 3
𝑥=𝑥

8. Restricting the Domain:
If a function is not one-to-one the domain can be restricted to portions that are one-to-one.

9. Restricting the Domain:
If a function is not one-to-one the domain can be restricted to portions that are one-to-one.
Increasing (−∞,−2)
Decreasing (−2,3)
Increasing (3,∞)
Has an inverse on each interval

10. Find the derivative of the inverse by implicit differentiation
( without solving for f -1 (x) )
Remember : f -1 (x) = f (y) ; therefore,
find
𝑦=2𝑥+sin⁡(𝑥)
𝑑𝑦 𝑑𝑦 =2 𝑑𝑥 𝑑𝑦 +cos⁡(𝑥) 𝑑𝑥 𝑑𝑦
1=(2+𝑐𝑜𝑠 𝑥 ) 𝑑𝑥 𝑑𝑦
1 2+cos⁡(𝑥) = 𝑑𝑥 𝑑𝑦

11. Derivative of the Inverse
f(a,b) =m
Derivative of the Inverse
(a,b)
The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals.
𝑓 −1 𝑏,𝑎 = 1 𝑚
(b,a)
f(x) a = 3
𝑓 −1 𝑥 1 3
Derivative of an Inverse Function:
Given f is a differentiable one-to-one function and f -1 is the inverse of f . If b belongs to the domain of f -1 and
f /(f(x) ≠ 0 , then f -1(b) exists and
= 1 𝑓′(𝑎)

12. Derivative of the Inverse
(a,b)
The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals.
(b,a)
Derivative of an Inverse Function:
If is the derivative of f,
Then is the derivative of f -1(b)
CAUTION:
Pay attention to the plug in value!!!

13. Find the derivative of f -1 at (16,4)
(4,16)
ILLUSTRATION:
(16,4)
Find the derivative of f -1 at (16,4)
a) Find the Inverse b) Use the formula.
𝑓 −1 𝑥 = (𝑥) 1 2
𝑦= 𝑥 2
𝑦 ′ =2𝑥
(𝑓 −1 )′(𝑥)= (𝑥) −1 2
𝑦 ′ = 1 2𝑥
(𝑓 −1 )′(𝑥)= 1 2 𝑥
𝑦 ′ = 1 2(4) = 1 8
(𝑓 −1 )′(𝑥)= = 1 8
𝑓 −1 ′ 𝑏 = 1 8

14. EX:
Find the derivative of the Inverse at the given point, (b,a).
6= 𝑥 3 +7
−1= 𝑥 3
(-1,6)
𝑓 ′ 𝑥 = 3𝑥 2
𝑓 ′ −1 =3
(𝑓 −1 )′(6)= 1 𝑓 ′ −1 = 1 3
Theorem:

15. x f f / 10 3 2 3 10 4 Inverse Functions 𝑠 ′ 3 = 1 𝑓′(10) = 1 2
REMEMBER:
The x in the inverse (S) is the y in the original (f)
𝑠 ′ 3 = 1 𝑓′(10) = 1 2
If S(x) = f -1 (x), then S / (3) =
If S(x) = f -1 (x), then S / (10) =
3 is the y value
𝑠 ′ 10 = 1 𝑓′(3) = 1 4
10 is the y value

16. Last Update
1/8/14
Assignment: Worksheet 91