Handout - Derivative - Chain Rule

Power-Chain Rule a,b are constants.

Function Derivative

dy y = a · xn = a · n · xn−1 Power Rule dx

dy du y = a · un = a · n · un−1 · Power-Chain Rule dx dx

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

Answer: y0 = 384(6x + 21)7 a = 8, n = 8 du u = 6x + 21 ⇒ dx = 6

⇒ y0 = 8 · 8 · (6x + 21)7 · 6

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

3 Answer: y0 = 32(8x + 7) 4x2 + 7x + 28
a = 8, n = 4 2 du u = 4x + 7x + 28 ⇒ dx = 8x + 7

⇒ y0 = 8 · 4 · 4x2 + 7x + 28
3 · (8x + 7)

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

12x + 4 Answer: y0 = √ 6x2 + 4x + 26 1 a = 2, n = 2 2 du u = 6x + 4x + 26 ⇒ dx = 12x + 4

⇒ y0 = 2 · 1 · √ 1 · (12x + 4) 2 6x2+4x+26

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

√ √ g) 5 x2 + 8x + 25 h)7 7x2 + 3x + 24

i) √ 8 j) √ 3 3x2+3x+22 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; 91 2
10/3 2
19/3 e) 3 (14x + 9) 7x + 9x + 24 ; f) 22(14x + 4) 7x + 4x + 29 ; 5(2x+8) 7(14x+3) g) √ ; h) √ ; 2 x2+8x+25 2 7x2+3x+24 i) − 4(6x+3) ; j) − 3(4x+2) ; (3x2+3x+22)3/2 2(2x2+2x+22)3/2 2 Sine and Cosine - Chain Rules a,b are constants.

Function Derivative

dy y = sin(x) = cos(x) Sine Rule dx

dy y = cos(x) = − sin(x) Cosine Rule dx

dy du y = a · sin(u) = a · cos(u) · Chain-Sine Rule dx dx

dy du y = a · cos(u) = −a · sin(u) · Chain-Cosine Rule dx dx

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

Answer: 2 27x2 + 6x
cos 9x3 + 3x2 + 1
a = 2 3 2 du 2 u = 9x + 3x + 1 ⇒ dx = 27x + 6x

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

Answer: −5 45x4 + 20x3
sin 9x5 + 5x4 + 3
a = 5 5 4 du 4 3 u = 9x + 5x + 3 ⇒ dx = 45x + 20x

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

2 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; 1 1 4 5
3 4

5
4

− 2 i) 2 · 125x cos 5x − 140x sin 7x · 5 sin 5x + 5 cos 7x ; 1 3 2 3
2

3
2

2 j) 2 · 36x cos 2x − 80x sin 8x · 6 sin 2x + 5 cos 8x ; 4 Exponent and Logarithmic - Chain Rules a,b are constants.

Function Derivative

dy y = ex = ex Exponential Function Rule dx

dy 1 y = ln(x) = Logarithmic Function Rule dx x

dy du y = a · eu = a · eu · Chain-Exponent Rule dx dx

dy a du y = a · ln(u) = · Chain-Log Rule dx u dx

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

Answer: y0 = 42e7x+22 a = 6 du 0 7x+22 u = 7x + 22 ⇒ dx = 7 ⇒ y = 6 · e · 7

2 Ex3b. Find the derivative of y = 6e7x +3x+22

2 Answer: y0 = 6 · (14x + 3) · e7x +3x+22 a = 6 2 du 0 7x2+3x+22
u = 7x + 3x + 22 ⇒ dx = 14x + 3 ⇒ y = 6 · e · 14x + 3

8/3 7/2 Ex3c. Find the derivative of y = −2e8x +7x

5/2 5/3 49x 64x 8/3 7/2 Answer: y0 = −2 · +
· e8x +7x 2 3 a = −2 8/3 7/2 du 49x5/2 64x5/3 u = 8x + 7x ⇒ dx = 2 + 3 0 8x8/3+7x7/2 49x5/2 64x5/3
⇒ y = −2 · e · 2 + 3

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

21 Answer: y0 = 3x + 26 a = 7 du 0 7 u = 3x + 26 ⇒ dx = 3 ⇒ y = 3x+26 · 3

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

8(4x + 9) Answer: y0 = 2x2 + 9x + 26 a = 8 2 du u = 2x + 9x + 26 ⇒ dx = 4x + 9 0 8
⇒ y = 2x2+9x+26 · 4x + 9

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

45x3/2 25x2/3  1 Answer: y0 = 7 · + · 2 3 9x5/2 + 5x5/3 a = 7 5/2 5/3 du 45x3/2 25x2/3 u = 9x + 5x ⇒ dx = 2 + 3 0 9x5/2+5x5/3 45x3/2 25x2/3
⇒ y = 7 · e · 2 + 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
h)6 ln 7x2 + 3x + 25

√ √ i) 6e5x8/3+5 x j)9 ln 9x8/3 + 8 x

8x+26 6x+21 8 18 Answers a) 48e ; b) 36e ; c) 2x+21 ; d) 3x+27 ; e) 6 · (6x + 8) · e3x2+8x+27; f) 8 · (8x + 6) · e4x2+6x+29; 4(8x+8) 6(14x+3) g) 4x2+8x+29 ; h) 7x2+3x+25 ;   5/3 8/3 √ i) 6 · 40x + √5
· e5x +5 x; j) 9 · 24x5/3 + √4 · 1 √ ; 3 2 x x 9x8/3+8 x 7 Exercises

Find the derivatives of the expressions

a) 5 sin(5x) + 4e4x+1 + √4 b) 2 cos(5x) + 5e3x+1 + √ 8 x x−1

c) 2 cos(4x) + 4 ln(3x − 1) − 7 d) cos(5x) + 5ex+1 − 7(x − 1)2/3 (3x+1)3/5

 √ 5  √ 3 e) 3 cos(5x) + 2 ln(4x) + 8 x f) cos(4x) + 3 ln(3x) − 7 5 x

1 5  − 2   2 √6 3x √4 g) 4 sin(3x) + 3 ln(x) − x h) 4 sin(5x) + e − 4 x

Answers a) 25 cos(5x) + 16e4x+1 − 2 ; b) −10 sin(5x) + 15e3x+1 − 4 ; x3/2 (x−1)3/2 c) −8 sin(4x) + 12 + 63 ; d) −5 sin(5x) + 5ex+1 − √14 ; 3x−1 5(3x+1)8/5 3 3 x−1    √ 4 2 √4 e) 5 · − 15 sin(5x) + x + x · 3 cos(5x) + 2 ln(4x) + 8 x ;    √ 2 f) 3 · − 4 sin(4x) + 3 − 7 · cos(4x) + 3 ln(3x) − 7 5 x ; x 5x4/5

3    − 2 g) − 1 · 12 cos(3x) + 3 + 3 · 4 sin(3x) + 3 ln(x) − √6 ; 2 x x3/2 x

3     2 h) 5 · 20 cos(5x) + 3e3x + 1 · 4 sin(5x) + e3x − √4 ; 2 x5/4 4 x 8