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2014 Derivatives of Inverse Functions
AP Calculus
InversesExistence 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
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.
Monotonic
– alway
s
increasi
ng or alw
ays
decrea
sing
𝑆𝑤𝑎𝑝 𝑥 𝑎𝑛𝑑 𝑦𝑦=𝑥2
𝑥=𝑦 2
±√𝑥=𝑦
Find the inverse𝑦=
𝑥+3𝑥+1
Switch x and y𝑥=𝑦+3𝑦+1
𝑥 (𝑦+1 )=𝑦+3
𝑥𝑦+𝑥=𝑦+3𝑥𝑦− 𝑦=3 −𝑥𝑦 (𝑥−1 )=3−𝑥
𝑦=3 −𝑥𝑥−1
multiply
distribute
Collect y
factor
divide
Find the inverse
𝑦=3√𝑥+4
𝑥=3√𝑦+4
𝑥3=𝑦+4
𝑥3− 4=𝑦
𝑦=(𝑥2− 4 ) for x ≥ 2 makes it monotonic
REVIEW: Inverse Functions
(a,b)
(b,a)
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)
Theorem:
f and g are inverses iff
f(g(x)) = g(f(x)) = x
To find f -1(x)
Reverse the x and y and resolve for y.
3
1
xy
x
𝑓 (𝑥 )=𝑥3 − 4 𝑔 (𝑥 )= 3√𝑥+4
𝑓 (𝑔 (𝑥 ) )=𝑔 ( 𝑓 (𝑥 ) )
( 3√𝑥+4 )3− 4=
3√𝑥3 − 4+4
𝑥+4 − 4=3√𝑥3
𝑥=𝑥
Restricting the Domain:If a function is not one-to-one the domain can be restricted to portions that are one-to-one.
x
y
3( ) 5 1f x x x
Restricting the Domain:If a function is not one-to-one the domain can be restricted to portions that are one-to-one.
x
y
Increasing (
Decreasing
Increasing (3,
Has an inverse on each interval
( ) 2 sin( )f x x x
Find the derivative of the inverse by implicit differentiation( without solving for f -1 (x) )
Remember : f -1 (x) = f (y) ; therefore,
find dx
dy
𝑦=2 𝑥+sin (𝑥 )𝑑𝑦𝑑 𝑦
=2𝑑𝑥𝑑𝑦
+cos (𝑥 )𝑑𝑥𝑑𝑦
1=(2+𝑐𝑜𝑠 (𝑥 )) 𝑑𝑥𝑑𝑦
12+cos (𝑥)
=𝑑𝑥𝑑𝑦
Derivative of the Inverse
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
(a,b)
(b,a)
The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals.
1
/ 1
1( )
( )f b
f f b
f(a,b) =m
𝑓 −1 (𝑏 ,𝑎 )= 1𝑚
f(x) slope @ a = 3
𝑓 −1 (𝑥 )𝑠𝑙𝑜𝑝𝑒@𝑏=13
¿1
𝑓 ′ (𝑎)
Derivative of the Inverse
Derivative of an Inverse Function:
If is the derivative of f,
Then is the derivative of f -1(b)
x a
dy
dx
1
x a
dydx
(a,b)
(b,a)
The SLOPES of the function and its inverse at the respective points (a,b) and (b,a) are reciprocals.
CAUTION:
Pay attention to the plug in value!!!
ILLUSTRATION:
2
1
( )
( )
f x x x o
f x x
Find the derivative of f -1 at (16,4)
2( )f x x
1( )f x x
a) Find the Inverse. b) Use the formula.
(4,16)
(16,4)
𝑓 −1 (𝑥 )=(𝑥 )12
( 𝑓 ¿¿− 1) ′ (𝑥)=12(𝑥)
− 12 ¿
( 𝑓 ¿¿− 1) ′ (𝑥)=1
2√𝑥¿
( 𝑓 ¿¿− 1) ′ (𝑥)=1
2√16=
18
¿
𝑦=𝑥2
𝑦 ′=2𝑥
𝑦 ′=1
2𝑥
( 𝑓 − 1 )′ (𝑏 )=18
𝑦 ′=1
2(4)=
18
EX:Find the derivative of the Inverse at the given point, (b,a).
3( ) 7 6f x x at b
Theorem: 1 1( )
( )f b
f a
6=𝑥3+7−1=𝑥3 (-1,6)
( 𝑓 ¿¿− 1) ′ (6)=1
𝑓 ′ (−1 )=
13
¿
𝑓 ′ (𝑥 )=3𝑥2
𝑓 ′ (− 1 )=3
Inverse Functions
x f f /
10 3 2
3 10 4
If S(x) = f -1 (x), then S / (3) =
If S(x) = f -1 (x), then S / (10) =
REMEMBER: The x in the inverse (S) is the y in the original (f)
3 is th
e y valu
e𝑠′ (3 )= 1
𝑓 ′ (10)=
12
𝑠′ (10 )= 1𝑓 ′(3)
=1410 is
the y
value
Last Update
• 1/8/14
• Assignment: Worksheet 91