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 β alw
ays
increasi
ng or always
decreas
ing
ππ€ππ π₯ πππ π¦π¦=π₯2
π₯=π¦ 2
Β±βπ₯=π¦
Find the inverseπ¦=
π₯+3π₯+1
Switch x and yπ₯=π¦+3π¦+1
π₯ (π¦+1 )=π¦+3π₯π¦+π₯=π¦+3π₯π¦β π¦=3 βπ₯π¦ (π₯β1 )=3 βπ₯
π¦=3 βπ₯π₯β1
multiply
distributeCollect y
factor
divide
Find the inverse
π¦=3βπ₯+4
π₯=3βπ¦+4
π₯3=π¦+4
π₯3β 4=π¦
π¦=(π₯2β 4 ) for x β₯ 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.
31
xyx
π (π₯ )=π₯3 β 4 π (π₯ )= 3βπ₯+4
π (π (π₯ ) )=π ( π (π₯ ) )
( 3βπ₯+4 )3β 4=
3βπ₯3 β 4+4
π₯+4 β 4=3βπ₯3
π₯=π₯
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 dxdy
π¦=2 π₯+sin (π₯ )ππ¦π π¦=2 ππ₯ππ¦ +cos (π₯ )
ππ₯ππ¦
1=(2+πππ (π₯ )) ππ₯ππ¦1
2+cos (π₯)=ππ₯ππ¦
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) β 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 bf f b
f(a,b) =m
π β1 (π ,π )= 1π
f(x) slope @ a = 3
π β1 (π₯ )π ππππ@π=13
ΒΏ1
π β² (π)
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
dydx
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)
π β1 (π₯ )=(π₯ )12
( π ΒΏΒΏβ 1) β² (π₯)=12(π₯)
β 12 ΒΏ
( π ΒΏΒΏβ 1) β² (π₯)=1
2βπ₯ΒΏ
( π ΒΏΒΏβ 1) β² (π₯)=1
2β16=
18 ΒΏ
π¦=π₯2
π¦ β²=2π₯π¦ β²=
12π₯
( π β 1 )β² (π )=18
π¦ β²=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 bf a
6=π₯3+7β1=π₯3 (-1,6)
( π ΒΏΒΏβ 1) β² (6)=1
π β² (β1 )=
13 ΒΏ
π β² (π₯ )=3π₯2
π β² (β 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 the y
value
π β² (3 )= 1π β² (10)
=12
π β² (10 )= 1π β²(3)
=1410 is
the y valu
e
Last Update
β’ 1/8/14
β’ Assignment: Worksheet 91
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