AP Calculus BCTuesday, 22 September 2015

• OBJECTIVE TSW apply the Chain Rule to differentiate functions.

• ASSIGNMENT DUE TOMORROW/THURSDAY– Sec. 3.4: pp. 170-171 (7, 8, 11, 12, 15-19

odd, 20, 25, 26, 40. 41)– Sec. 3.3: p. 161 (44, 45, 47)

• TODAY’S ASSIGNMENT (due Friday, 09/25/15)

– Sec. 3.7: pp. 198-199 (7-10 all, 19-33 odd (omit 27), 57, 61, 65)

Sec. 3.7: The Chain Rule

Sec. 3.7: The Chain Rule Easy to differentiate:

Not easy to differentiate:

Easy to differentiate:

Not easy to differentiate:

Easy to differentiate:

Not easy to differentiate:

3 4y x

233 4y x

cosf x x

cos7f x x

9 7g x x

9 7g x x

Sec. 3.7: The Chain Rule

How does one item change with respect to another?

Ex: There are 3 gears, labeled #1, #2, and #3. Gear one is the smallest, followed by gear two, and finally gear three.

#1 rotates four times as fast as #2. #2 rotates five times as fast as #3.

How much faster is #1 rotating than #3?

(4)(5) = 20 times as fast

This demonstrates the CHAIN RULE.

Sec. 3.7: The Chain Rule

The Chain Rule

Sec. 3.7: The Chain Rule Ex: Differentiate:

121a) f x x

23b) 2f x x

111 112f x x 11

12 1x

1 223 2x

1 221

3 22

6xf x x

2

3

3 2

x

x

Sec. 3.7: The Chain Rule

The General Power Rule

Sec. 3.7: The Chain Rule Ex: Differentiate:

1a)

3 1f x

x

13 1x

21 3 1 3f x x

23

3 1x

Sec. 3.7: The Chain Rule Ex: Differentiate:

cos 3b) y x

sin 33y x

3 sin 3 x

Sec. 3.7: The Chain Rule Ex: Differentiate:

2c tc) oy x 2cot x

22 c csot cy x x

22cot cscx x

Sec. 3.7: The Chain Rule Ex: Find all points on the graph of

223 1f x x for which 0 andf x

DNE.f x

2 32 1f x x

1 32 1 2

2

3f x x x

3 2

4

3 1

x

x

Sec. 3.7: The Chain Rule Ex: Find all points on the graph of

223 1f x x for which 0 andf x

DNE.f x

3 2

4

3 1

xf x

x

0 :f x 4 0x 0x 0,1

DNE :f x 3 23 1 0x 1x 1, 0 , 1, 0

Sec. 3.7: The Chain Rule Ex: Differentiate:

s) n4a iy x

2cos 3b) 2f x x

cos4 4xy 4cos4x

2sin 3 62x xf x

26 sin 3 2x x

Sec. 3.7: The Chain Rule Ex: Differentiate:

3t nc) ay x

3tand) g x x

3tan x

2 3 2sec 3y x x 2 2 33 secx x

3tanx

2 23 tan secg xx x 2 23 tan secx x

Sec. 3.7: The Chain Rule Ex: Differentiate:

2 3coe) s 4y x 23cos 4x

3 3 212si2 cos 44 n xxy x Double - angle for sine :

2sin cos sin2 2 3 32sin o2 4 s 41 cx xx

2 312 sin 8x x

2 312 sin 8x x

Sec. 3.7: The Chain Rule Ex: Differentiate:

3 22a) f x x x 1 23 22x x

1 2 1 23 2 2 2122 2 3

2f x x xx x x

1 2 1 24 2 2 22 3 2x x x x

1 2 12 2 2 22 3 2x x x x

2 2 2

1 22

6 3

2

x x x

x

2 2

1 22

4 6

2

x x

x

2 2

2

2 2 3

2

x x

x

Sec. 3.7: The Chain Rule Ex: Differentiate:

3 2

b)4

xf x

x

1 32 4

x

x

1 3 2 32 2

21 32

14 1 4

32

4

x x xf x

x

x

2 3 12 2 2

2 32

24 4

3

4

x x x

x

2

4 32

14

3

4

x

x

2

4 32

12

3 4

x

x

3

3

Sec. 3.7: The Chain Rule Ex: Differentiate:

3

2c)

2 1

4

xy

x

2 22

2 2 4 22 13

4

2 1 2

4

x x x

x

xy

x

2 22

2 22

2 8 4 22 13

4 4

x x xx

x x

2 2

2 22

2 1 2 2 83

4 4

x x x

x x

22

2 22

2 42 13

4 4

x xx

x x

2 2

42

6 2 1 4

4

x x x

x

Sec. 3.7: The Chain Rule

Summary of Differentiation Rules

Sec. 3.7: The Chain Rule PPT Problems (put on a clean sheet of

notebook paper): Due tomorrow/Thursday, 09/23-24/15.

)2

11

f xx

2

2

3

3)

12

xf x

x

2co3) s 3y x 2si 34 n) y x

Find each derivative; simplify your answer.