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