Handout - Derivative, Lecture notes of Calculus

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Handout - Derivative - Chain Rule & Sin(x), Cos(x), ex, ln(x)

Power-Chain Rule a,b are constants.

Function Derivative

y = a · xn^

dy

dx

= a · n · xn−^1 Power Rule

y = a · un^

dy

dx

= a · n · un−^1 ·

du

dx

Power-Chain Rule

Exercises

Find the derivatives of the expressions

a) 6(9x + 26)^9 b) 6(8x + 29)^8

c) 5

x^2 + 3x + 23

d)

7 x^2 + 2x + 22

e) 7

2 x^2 + 5x + 28

f)

5 x^2 + 7x + 23

g) 4

4 x^2 + 3x + 26 h)

2 x^2 + 5x + 27

i) √^9 3 x^2 +6x+

j) √^9 5 x^2 +6x+

Answers a) 486(9x + 26)^8 ; b) 384(8x + 29)^7 ;

c) 45(2x + 3)

x^2 + 3x + 23

; d) 30(14x + 2)

7 x^2 + 2x + 22

e) 1753 (4x + 5)

2 x^2 + 5x + 28

; f) 1283 (10x + 7)

5 x^2 + 7x + 23

g) √2(8x+3) 4 x^2 +3x+

; h) √3(4x+5) 2 x^2 +5x+

i) −

9(6x+6) 2(3x^2 +6x+29)^3 /^2

; j) −

9(10x+6) 2(5x^2 +6x+25)^3 /^2

Exercises

Find the derivatives of the expressions

a) 7 cos

2 x^2 + 5x + 9

b) 8 cos

5 x^3 + 5x^2 + 2

c) 5 cos

x^3 + 3x^2 + 1

d) 7 cos

7 x^2 + 1

e)

6 sin

4 x^4

• 4 cos(3x)

f)

7 sin

9 x^2

• 5 cos

6 x^5

g)

8 sin

6 x^4

• cos

8 x^4

h)

4 sin

8 x^4

• cos

8 x^2

i)

7 sin(4x) + 6 cos

6 x^4

(^2) j)

7 sin(5x) + 9 cos

4 x^3

2

Answers a) −7(4x + 5) sin

2 x^2 + 5x + 9

; b) − 8

15 x^2 + 10x

sin

5 x^3 + 5x^2 + 2

c) − 5

3 x^2 + 6x

sin

x^3 + 3x^2 + 1

; d) − 98 x sin

7 x^2 + 1

e) 3 ·

96 x^3 cos

4 x^4

− 12 sin(3x)

6 sin

4 x^4

• 4 cos(3x)

f) 4 ·

126 x cos

9 x^2

− 150 x^4 sin

6 x^5

7 sin

9 x^2

• 5 cos

6 x^5

g) 3 ·

192 x^3 cos

6 x^4

− 32 x^3 sin

8 x^4

8 sin

6 x^4

• cos

8 x^4

h) 4 ·

128 x^3 cos

8 x^4

− 16 x sin

8 x^2

4 sin

8 x^4

• cos

8 x^2

i) 52 ·

28 cos(4x) − 144 x^3 sin

6 x^4

7 sin(4x) + 6 cos

6 x^4

j) 12 ·

35 cos(5x) − 108 x^2 sin

4 x^3

7 sin(5x) + 9 cos

4 x^3

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

Function Derivative

y = ex^

dy

dx

= ex^ Exponential Function Rule

y = ln(x)

dy

dx

x

Logarithmic Function Rule

y = a · e

u dy dx

= a · e

u ·

du

dx

Chain-Exponent Rule

y = a · ln(u)

dy

dx

a

u

du

dx

Chain-Log Rule

Exercises

Find the derivatives of the expressions

a) 5 cos(4x) + 3e^3 x+1^ − 2

2 x − 1 b) 4 sin(3x) + 2ex+1^ + 4

2 x − 1

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

4 (x−1)^5 /^8

e)

sin(4x) + 2e^4 x^ − √ (^35) x

f)

5 sin(4x) + 2 ln(3x) + √^6 x

g)

4 sin(2x) + e^5 x^ + √ (^54) x

2 h)

sin(4x) + 2 ln(5x) + 4 7

x

2

Answers a) −20 sin(4x) + 9e^3 x+1^ − 4 5(2x−1)^4 /^5

; b) 12 cos(3x) + 2ex+1^ + 8 5(2x−1)^4 /^5

c) −8 sin(4x) + (^4) x^12 +1 + 40 3 3

√ 4 x− 1

; d) 15 cos(3x) + 5ex−^1 + 5 2(x−1)^13 /^8

e) 3 ·

4 cos(4x) + 8e^4 x^ + 5 3 x^4 /^3

sin(4x) + 2e^4 x^ − √ 35 x

f) 3 ·

20 cos(4x) +

2 x −^

3 x^3 /^2

5 sin(4x) + 2 ln(3x) + √^6 x

g)

1 2 ·

8 cos(2x) + 5e^5 x^ −

4 5 x^6 /^5

4 sin(2x) + e^5 x^ +

4 √ (^5) x

2 ;

h) −

1 2 ·

4 cos(4x) +

2 x +^

4 7 x^6 /^7

sin(4x) + 2 ln(5x) + 4 7

x

2 ;

Practice - Chain Rule

P1. Find the derivative of y = 7

4 x^2 + 7x + 22

P2. Find the derivative of y =

x^2 + 13x + 25

P3. Find the derivative of y = 6(8x − 7)^9 +

(2x + 9)^4

P4. Find the derivative of y = 2

3 x

2

• 6

(8x^3 − 1)

19 / 2

P5. Find the derivative of y = 4e

6 x^5 +9x+

P6. Find the derivative of y = 7 ln

9 x^5 + 3x + 25

P7. Find the derivative of y =

2 sin (6x^4 ) + 2 cos (9x^3 )

P8. Find the derivative of y = 5 sin(5x) + 5 ln(3x + 1) + 7(x − 1)^7

ANSWERS: P1) 56(8x+7)

4 x^2 + 7x + 22

P2) −

4(2x+13) (x^2 +13x+25)^3 /^2

P3) 432(8x−7)^8 − (^) (2x^16 +9) 5

. P4)

8 x 5(3x^2 +6)^13 /^15

1140 x^2 (8x^3 −1)^21 /^2

P5) 4 · (30x^4 + 9) · e^6 x

(^5) +9x+ P6)

7 ( 45 x^4 +3) 9 x^5 +3x+

. P7) −

48 x^3 cos( 6 x^4 )− 54 x^2 sin( 9 x^3 ) 2(2 sin(6x^4 )+2 cos(9x^3 ))^3 /^2

. P8) 25 cos(5x) + (^3) x^15 +1 + 49(x − 1)^6