Handout - Derivative - Chain Rule Power-Chain Rule

Handout - Derivative - Chain Rule

Power-Chain Rule a,b are constants.

Function

Derivative

y = a ? xn

dy = a ? n ? xn-1 dx

y = a ? un

dy = a ? n ? un-1 ? du

dx

dx

Power Rule Power-Chain Rule

Ex1a. Find the derivative of y = 8(6x + 21)8

Answer: y = 384(6x + 21)7

a = 8, n = 8

u = 6x + 21

du dx

=

6

y = 8 ? 8 ? (6x + 21)7 ? 6

Ex1b. Find the derivative of y = 8 4x2 + 7x + 28 4

Answer: y = 32(8x + 7) 4x2 + 7x + 28 3

a = 8, n = 4

u = 4x2 + 7x + 28

du dx

=

8x

+

7

y = 8 ? 4 ? 4x2 + 7x + 28 3 ? (8x + 7)

Ex1c. Find the derivative of y = 2 6x2 + 4x + 26

12x + 4 Answer: y =

6x2 + 4x + 26

a = 2,

n

=

1 2

u = 6x2 + 4x + 26

du dx

=

12x

+

4

y

=

2

?

1 2

?

1 6x2+4x+26

?

(12x

+

4)

1

Exercises Find the derivatives of the expressions

a) 5(9x + 25)8

b) 7(2x + 24)8

c) 2 4x2 + 4x + 21 9

d)6 7x2 + 4x + 22 4

e) 7 7x2 + 9x + 24 13/3

g) 5 x2 + 8x + 25

f)3 7x2 + 4x + 29 22/3

h)7 7x2 + 3x + 24

i)

8 3x2+3x+22

j)

3 2x2+2x+22

Answers a) 360(9x + 25)7; b) 112(2x + 24)7; c) 18(8x + 4) 4x2 + 4x + 21 8; d) 24(14x + 4) 7x2 + 4x + 22 3;

e)

91 3

(14x

+

9)

7x2 + 9x + 24

10/3;

f) 22(14x + 4) 7x2 + 4x + 29 19/3;

g)

5(2x+8) 2 x2+8x+25

;

h)

7(14x+3) 2 7x2+3x+24

;

i)

-

4(6x+3) (3x2+3x+22)3/2

;

j)

-

3(4x+2) 2(2x2+2x+22)3/2

;

2

Sine and Cosine - Chain Rules a,b are constants.

Function y = sin(x) y = cos(x) y = a ? sin(u) y = a ? cos(u)

Derivative dy

= cos(x) dx

dy = - sin(x)

dx

dy

du

= a ? cos(u) ?

dx

dx

dy

du

= -a ? sin(u) ?

dx

dx

Sine Rule Cosine Rule Chain-Sine Rule Chain-Cosine Rule

Ex2a. Find dy where y = 2 sin 9x3 + 3x2 + 1 dx

Answer: 2 27x2 + 6x cos 9x3 + 3x2 + 1

a=2

u = 9x3 + 3x2 + 1

du dx

=

27x2

+ 6x

Ex2b. Find dy where y = 5 cos 9x5 + 5x4 + 3 dx

Answer: -5 45x4 + 20x3 sin 9x5 + 5x4 + 3

a=5

u = 9x5 + 5x4 + 3

du dx

=

45x4

+ 20x3

Ex2c. Find the derivative of y = 4 sin(4x) + 4 cos(5x) 4

Answer: 4 ? 16 cos(4x) - 20 sin(5x) ? 4 sin(4x) + 4 cos(5x) 3

n=4

u = 4 sin(4x) + 4 cos(5x)

du dx

=

16 cos(4x)

-

20 sin(5x)

3

Exercises Find the derivatives of the expressions

a) 7 sin 7x3 + 7x2 + 8

b) 5 cos 2x4 + 7x3 + 9

c) 8 sin 8x3 + 9x2 + 1

d) 5 cos 6x4 + 8x3 + 2

e) sin(x) + 9 cos x4 4

f) 2 sin(6x) + 9 cos 3x2 3

g) 4 sin 5x3 + 8 cos 8x5 5

h) sin 7x3 + 2 cos 6x4 3

1

i) 5 sin 5x5 + 5 cos 7x4 2

3

j) 6 sin 2x3 + 5 cos 8x2 2

Answers a) 7 21x2 + 14x cos 7x3 + 7x2 + 8 ; b) -5 8x3 + 21x2 sin 2x4 + 7x3 + 9 ;

c) 8 24x2 + 18x cos 8x3 + 9x2 + 1 ; d) -5 24x3 + 24x2 sin 6x4 + 8x3 + 2 ;

e) 4 ? cos(x) - 36x3 sin x4 ? sin(x) + 9 cos x4 3;

f) 3 ? 12 cos(6x) - 54x sin 3x2 ? 2 sin(6x) + 9 cos 3x2 2;

g) 5 ? 60x2 cos 5x3 - 320x4 sin 8x5 ? 4 sin 5x3 + 8 cos 8x5 4;

h) 3 ? 21x2 cos 7x3 - 48x3 sin 6x4 ? sin 7x3 + 2 cos 6x4 2;

i)

1 2

?

125x4 cos

5x5

- 140x3 sin

7x4

? 5 sin 5x5 + 5 cos 7x4

-

1 2

;

j)

3 2

?

36x2 cos

2x3

- 80x sin

8x2

? 6 sin 2x3 + 5 cos 8x2

1

2;

4

Exponent and Logarithmic - Chain Rules a,b are constants.

Function y = ex y = ln(x) y = a ? eu y = a ? ln(u)

Derivative dy = ex dx

dy 1 =

dx x

dy = a ? eu ? du

dx

dx

dy a du =?

dx u dx

Exponential Function Rule Logarithmic Function Rule Chain-Exponent Rule Chain-Log Rule

Ex3a. Find the derivative of y = 6e7x+22

Answer: y = 42e7x+22 a=6 u = 7x + 22

du dx

=

7

y

= 6 ? e7x+22 ? 7

Ex3b. Find the derivative of y = 6e7x2+3x+22

Answer: y = 6 ? (14x + 3) ? e7x2+3x+22

a=6

u = 7x2 + 3x + 22

du dx

=

14x + 3

y

= 6 ? e7x2+3x+22 ?

14x + 3

Ex3c. Find the derivative of y = -2e8x8/3+7x7/2

Answer: y = -2 ?

49x5/2 64x5/3 +

? e8x8/3+7x7/2

2

3

a = -2

u = 8x8/3 + 7x7/2

du dx

=

49x5/2 2

+

64x5/3 3

y = -2 ? e8x8/3+7x7/2 ?

49x5/2 2

+

64x5/3 3

5

Ex3d. Find the derivative of y = 7 ln(3x + 26)

21 Answer: y =

3x + 26 a=7 u = 3x + 26

du dx

=

3

y

=

7 3x+26

?

3

Ex3e. Find the derivative of y = 8 ln 2x2 + 9x + 26

8(4x + 9) Answer: y =

2x2 + 9x + 26

a=8

u = 2x2 + 9x + 26

du dx

=

4x +

9

y

=

8 2x2+9x+26

?

4x + 9

Ex3f. Find the derivative of y = 7 log 9x5/2 + 5x5/3

45x3/2 25x2/3

1

Answer: y = 7 ?

+

?

2

3

9x5/2 + 5x5/3

a=7

u = 9x5/2 + 5x5/3

du dx

=

45x3/2 2

+

25x2/3 3

y = 7 ? e9x5/2+5x5/3 ?

45x3/2 2

+

25x2/3 3

6

Exercises Find the derivatives of the expressions

a) 6e8x+26

b) 6e6x+21

c) 4 ln(2x + 21)

d) 6 ln(3x + 27)

e) 6e3x2+8x+27

f )8e4x2 +6x+29

g) 4 ln 4x2 + 8x + 29

i)

6e5x8/3

+5 x

h)6 ln 7x2 + 3x + 25

j)9 ln

9x8/3

+

8x

Answers a) 48e8x+26;

b) 36e6x+21;

c)

8 2x+21

;

d)

18 3x+27

;

e) 6 ? (6x + 8) ? e3x2+8x+27; f) 8 ? (8x + 6) ? e4x2+6x+29;

g)

4(8x+8) 4x2+8x+29

;

h)

6(14x+3) 7x2+3x+25

;

i) 6 ?

40x5/3 3

+

5 2x

? e5x8/3+5x;

j) 9 ?

24x5/3

+

4 x

? 9x8/31+8x ;

7

Exercises Find the derivatives of the expressions

a)

5 sin(5x)

+

4e4x+1

+

4 x

b) 2 cos(5x) + 5e3x+1 + 8

x-1

c)

2 cos(4x)

+

4 ln(3x

-

1)

-

7 (3x+1)3/5

d) cos(5x) + 5ex+1 - 7(x - 1)2/3

5 e) 3 cos(5x) + 2 ln(4x) + 8 x

3 f) cos(4x) + 3 ln(3x) - 7 5 x

g)

4

sin(3x)

+

3

ln(x)

-

6 x

-

1 2

5

h) 4 sin(5x) + e3x - 44x 2

Answers

a)

25

cos(5x)

+

16e4x+1

-

2 x3/2

;

b)

-10

sin(5x)

+

15e3x+1

-

4 (x-1)3/2

;

c)

-8 sin(4x)

+

12 3x-1

+

63 5(3x+1)8/5

;

d) -5 sin(5x) + 5ex+1 - 3 31x4-1 ;

e) 5 ?

-

15

sin(5x)

+

2 x

+

4 x

4 ? 3 cos(5x) + 2 ln(4x) + 8 x ;

f) 3 ?

-

4 sin(4x)

+

3 x

-

7 5x4/5

?

2 cos(4x) + 3 ln(3x) - 7 5 x ;

-

3 2

g)

-

1 2

?

12

cos(3x)

+

3 x

+

3 x3/2

?

4

sin(3x)

+

3

ln(x)

-

6 x

;

3

h)

5 2

?

20

cos(5x)

+

3e3x

+

1 x5/4

?

4 sin(5x) + e3x - 44x

2

;

8

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