2010 – Chain Rule

AP CALCULUS

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Activity: Teacher-Directed Instruction

COMPOSITE FUNCTIONS

Know: Need:

Know: Need:  REM:

2( 2)y x 8( 2)y x 2( 2)y x 4 3 2 2(4 3 2 7 5)y x x x x

2010

10

t(x)=1.0825(x)

d(x)=.5(x)10.83

𝑓 (𝑥 )=𝑡(𝑑 (𝑥 ))

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Order matters

𝑑 (𝑥 ) h𝑤 𝑎𝑡 𝑖𝑠 𝑑𝑜𝑛𝑒 𝑓𝑖𝑟𝑠𝑡

COMPOSITE FUNCTIONS

( ( ))y f g x Let and

y = outside u = inside

function function

( )u g x( )y f u

g(x) - what is done first.

f(x)

xg(x)

g(x)

f(g(x))

COMPOSITE FUNCTIONS

DECOMPOSE y = (outside) u = (inside).

2 3( 1)y x

23 1y x x

1

1y

x

tan 2y x2tany x

2tany x

𝑦=𝑢3 𝑢=𝑥2+1

𝑦=𝑢12 𝑢=3 𝑥2 −𝑥+1

𝑦=𝑢− 1 𝑢=𝑥+1

𝑦=tan(𝑢) 𝑢=2𝑥

𝑦=tan(𝑢) 𝑢=𝑥2

𝑦=𝑢2 𝑢=tan(𝑥)

tan𝑥2

¿

( ( ( ))d

f g xdx

In Words:

__________________________________________________________

 Derivative of __________________________________________________

a composite function2

func

tions

2 pa

rts ¿ 𝑓 ′ (𝑔 (𝑥 ) )∗𝑔 ′ (𝑥)

d(outside)leave the inside alone*d(inside)

Texas two step

Chain Rule

( )u g x

Leibniz Notation:

In Words:

__________________________________________________________

( ( ))y f g x ( )y f u

𝑑𝑦𝑑𝑥

=𝑑𝑦𝑑𝑢

∗𝑑𝑢𝑑𝑥

d(outside)leave the inside alone*d(inside)

Example 1:

Example:

 

Let y = u =

 

  _______

dy

du

du

dx

dy

dx

2 3(3 2 1)y x x

𝑢3 3 𝑥2+2𝑥+1

3𝑢2 (6 𝑥+2)

3𝑢2∗(6 𝑥+2)

𝑑𝑦𝑑𝑥

=3(3𝑥¿¿2+2𝑥+1)2∗(6 𝑥+2)¿

Example 2:2 3(3 2 )y x x

Two steps: 𝑦=𝑢3 𝑢=3 𝑥−2 𝑥2

𝑦 ′=3 (3 𝑥− 2𝑥2 ) 2∗(3 − 4 𝑥)

𝑦 ′=(9−12 𝑥 )∗(3 𝑥− 2𝑥2)2

Example 3:

23 ( 2)y x 𝑦=(𝑥2+2)13

𝑦=𝑢13 𝑢=(𝑥2+2)2

𝑦 ′=13(𝑥2+2)

− 23 ∗(2 𝑥)

𝑦 ′=(23𝑥)¿

Example 4:

Example:1

(2 3)y

t

2

7

(3 5)y

x

Note: 2

1

4(3 5)y

x

¿ (2 𝑡−3)−1

𝑦=𝑢− 1 𝑢=(2𝑡− 3)

𝑦 ′=−1 (2 𝑡−3 )− 2∗(2)

𝑦 ′=−2(2𝑡− 3)−2

𝑦 ′=− 2

(2 𝑡−3)2

2

7

(3 5)y

x

𝑦=−7 ¿

𝑦 ′=14 (3 𝑥−5)− 3∗(3)

𝑦 ′=42 (3 𝑥−5)− 3

𝑦 ′=42

(3 𝑥− 5)2

𝑦=−7𝑢−2 𝑢=3 𝑥−5

2

1

4(3 5)y

x

𝑦=

14

(3 𝑥−5)− 2

𝑦=14𝑢− 2 𝑢=3 𝑥−5

𝑦 ′=− 24

(3 𝑥−5)− 3∗(3)

𝑦 ′=− 64

(3 𝑥− 5)−3

𝑦 ′=−3

2(3 𝑥−5)3

Example 6:

cos(3 )y x

2cosy x

𝑦 ′=−sin (3 𝑥 )∗3

𝑦 ′=−3 sin (3 𝑥)

𝑦=cos𝑢 𝑢=3 𝑥

𝑦=𝑢2 𝑢=cos 𝑥𝑦 ′=2¿

𝑦 ′=−2 sin (𝑥 )∗ cos (𝑥)

Example 7:Extended Chain:

Ex: OR

WORDS:

Extended Chain: ___________

d(outside)*d(middle)*d(inside)

¿¿𝑦 ′=3¿¿

𝑦 ′=6 𝑥 𝑠𝑖𝑛2(𝑥¿¿2)cos (𝑥¿¿ 2)¿¿

Number of functions = number of parts

𝑦=𝑠𝑖𝑛3 (𝑥¿¿ 2)¿

Derivative of the Absolute Value Function

REM:2x x

2d dx x

dx dx Do not simplify.

Use the Chain Rule.¿ (𝑥¿¿2)

12 ¿

𝑦 ′=12(𝑥2)

12∗(2 𝑥)

𝑦 ′=𝑥√𝑥

=𝑥

|𝑥|

𝑦=|𝑢|

𝑦 ′=𝑢|𝑢|

∗𝑢 ′

General Rules: Working with number values

Find the derivative.

1) f(x) + g(x) at x = 3 2) 2f(x) – 3g(x) at x =2

3) f(x)*g(x) at x = 2 4) f(x) / g(x) at x = 3

5) f(g(x)) at x = 2 6) (f(x))3 at x = 3

x f(x) g(x) f / (x) g / (x)

2 8 3 -4

3 2 -6 -512

23

𝑓 ′ (𝑥 )+𝑔 ′ (𝑥) = 2 𝑓 ′ (𝑥 )− 3𝑔 ′ (𝑥)¿2 (− 4 )− 3( 23 )=− 10

𝑓 𝑔′+𝑔𝑓 ′ 8 ( 23 )+3(− 4 ) 𝑔 𝑓 ′ − 𝑓𝑔′

𝑔2

(−6 )( 12 )−(2)(−5)

(−6)2

𝑓 ′ (𝑔 (𝑥 ) )∗𝑔 ′ (𝑥) 𝑓 ′ (𝑔 (2 ) )∗𝑔 ′ (2)

𝑓 ′ (3 )∗( 23 )=( 1

2 )( 23 )=1

3

𝑑𝑦𝑑𝑥

3

𝑓 (𝑥 )∗ 𝑓 ′(𝑥 )

3¿3∗22∗

12=3∗4∗

12=6

Last Update

• 10/13/07

• Assignment p. 153 # 13 – 31 odd, 56