Trigonometric Identities

A. Reciprocal Identities
1 1
1. 𝑠𝑖𝑛𝑥 = 𝑐𝑠𝑐𝑥 4. 𝑐𝑠𝑐𝑥 = 𝑠𝑖𝑛𝑥
1 1
2. 𝑐𝑜𝑠𝑥 = 𝑠𝑒𝑐𝑥 5. 𝑠𝑒𝑐𝑥 = 𝑐𝑜𝑠𝑥
1 1
3. 𝑡𝑎𝑛𝑥 = 𝑐𝑜𝑡𝑥 6. 𝑐𝑜𝑡𝑥 = 𝑡𝑎𝑛𝑥

B. Sum and Difference of Two Angles

1. sin(𝑥 ± 𝑦) = 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑦 ± 𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑦
2. cos(𝑥 ± 𝑦) = 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 ∓ 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦
𝑡𝑎𝑛𝑥 ±𝑡𝑎𝑛𝑦
3. tan(𝑥 ± 𝑦) = 1∓𝑡𝑎𝑛𝑥𝑡𝑎𝑛𝑦

C. Double Angle Identities

1. sin 2𝑥 = 2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥
2. 𝑐𝑜𝑠2𝑥 = cos 2 𝑥 − sin2 𝑥
3. 𝑐𝑜𝑠2𝑥 = 1 − 2 sin2 𝑥
4. 𝑐𝑜𝑠2𝑥 = 2 cos 2 𝑥 − 1
2𝑡𝑎𝑛𝑥
5. 𝑡𝑎𝑛2𝑥 = 1−tan2 𝑥

D. Pythagorean Identities
1. sin2 𝑥 + cos 2 𝑥 = 1
2. 1 + tan2 𝑥 = sec 2 𝑥
3. 1 + cot 2 𝑥 = csc 2 𝑥
Derivatives of Trigonometric Functions

𝑑 𝑑𝑢 𝑑 𝑑𝑢
(𝑠𝑖𝑛𝑢) = 𝑐𝑜𝑠𝑢 (𝑐𝑜𝑡𝑢) = −𝑐𝑠𝑐 2 𝑢
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥

𝑑 𝑑𝑢 𝑑 𝑑𝑢
(𝑐𝑜𝑠𝑢) = −𝑠𝑖𝑛𝑢 (𝑠𝑒𝑐𝑢) = 𝑠𝑒𝑐𝑢𝑡𝑎𝑛𝑢
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥

𝑑 𝑑𝑢 𝑑 𝑑𝑢
(𝑡𝑎𝑛𝑢) = 𝑠𝑒𝑐 2 𝑢 (𝑐𝑠𝑐𝑢) = −𝑐𝑠𝑐𝑢𝑐𝑜𝑡𝑢
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥
Find the derivative of each of the following functions:
1. 𝑓(𝑥 ) = 4 cos 𝑥 + 2𝑠𝑖𝑛𝑥
𝑓 ′ (𝑥 ) = 4(−𝑠𝑖𝑛𝑥 ) + 2(𝑐𝑜𝑠𝑥)
𝑓 ′ (𝑥 ) = −4𝑠𝑖𝑛𝑥 + 2𝑐𝑜𝑠𝑥

2. 𝑔(𝑥 ) = −4𝑥 2 𝑐𝑜𝑠𝑥 (𝑢𝑠𝑖𝑛𝑔 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑟𝑢𝑙𝑒)

𝑔′ (𝑥 ) = −8𝑥𝑐𝑜𝑠𝑥 − 4𝑥 2 (−𝑠𝑖𝑛𝑥 )
𝑔′ (𝑥 ) = −8𝑥𝑐𝑜𝑠𝑥 + 4𝑥 2 𝑠𝑖𝑛𝑥

5−𝑐𝑜𝑠𝑥
3. ℎ(𝑥 ) = 5+𝑠𝑖𝑛𝑥
(𝑢𝑠𝑖𝑛𝑔 𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡 𝑟𝑢𝑙𝑒)

′(
(5 + 𝑠𝑖𝑛𝑥)(−(−𝑠𝑖𝑛𝑥 ) − (5 − 𝑐𝑜𝑠𝑥)(𝑐𝑜𝑠𝑥)
ℎ 𝑥) =
(5 + 𝑠𝑖𝑛𝑥 )2

′(
5𝑠𝑖𝑛𝑥 + 𝑠𝑖𝑛2 𝑥 − 5𝑐𝑜𝑠𝑥 + 𝑐𝑜𝑠 2 𝑥
ℎ 𝑥) =
(5 + 𝑠𝑖𝑛𝑥)2

′( (𝑠𝑖𝑛2 𝑥+𝑐𝑜𝑠 2 𝑥)+5𝑠𝑖𝑛𝑥−5𝑐𝑜𝑠𝑥

ℎ 𝑥) = (𝑆𝑞𝑢𝑎𝑟𝑒𝑑 𝐼𝑑𝑒𝑛𝑡𝑖𝑡𝑦)
(5+𝑠𝑖𝑛𝑥)2

1 + 5𝑠𝑖𝑛𝑥 − 5𝑐𝑜𝑠𝑥
ℎ′ (𝑥 ) =
(5 + 𝑠𝑖𝑛𝑥)2
4. 𝑓(𝑥 ) = 𝑠𝑒𝑐𝑥 − √2𝑡𝑎𝑛𝑥
𝑓 ′ (𝑥 ) = (𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 ) − (√2𝑠𝑒𝑐 2 𝑥 + 0)
𝑓 ′ (𝑥 ) = 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 − √2𝑠𝑒𝑐 2 𝑥

5. 𝑔(𝑥 ) = 4𝑐𝑠𝑐𝑥 − 𝑐𝑜𝑡𝑥

𝑔′ (𝑥 ) = 4(−𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥 ) − (−𝑐𝑠𝑐 2 𝑥 )
𝑔′ (𝑥 ) = −4𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥 + 𝑐𝑠𝑐 2 𝑥

6. ℎ(𝑥 ) = 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 (𝑢𝑠𝑖𝑛𝑔 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑟𝑢𝑙𝑒)

ℎ′ (𝑥 ) = (𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥 )(𝑡𝑎𝑛𝑥 ) + (𝑠𝑒𝑐𝑥)(𝑠𝑒𝑐 2 𝑥)
ℎ′ (𝑥 ) = 𝑠𝑒𝑐𝑥𝑡𝑎𝑛2 𝑥 + 𝑠𝑒𝑐 3 𝑥

𝑐𝑜𝑡𝑥
7. 𝑓(𝑥 ) = 1+𝑐𝑠𝑐𝑥 (𝑢𝑠𝑖𝑛𝑔 𝑐ℎ𝑎𝑖𝑛 𝑟𝑢𝑙𝑒)
2
(1 + 𝑐𝑠𝑐𝑥)(− csc 𝑥) − (𝑐𝑜𝑡𝑥)(−𝑐𝑜𝑡𝑥𝑐𝑠𝑐𝑥)
𝑓 ′ (𝑥 ) =
(1 + 𝑐𝑠𝑐𝑥 )2
2 2
[(1 + 𝑐𝑠𝑐𝑥)(− csc 𝑥) + (cot 𝑥𝑐𝑠𝑐𝑥)]
𝑓 ′ (𝑥 ) =
(1 + 𝑐𝑠𝑐𝑥 )2
( )( ) 2
−𝑐𝑠𝑐𝑥 [ 1 + 𝑐𝑠𝑐𝑥 𝑐𝑠𝑐𝑥 − (cot 𝑥)]
𝑓 ′ (𝑥 ) =
(1 + 𝑐𝑠𝑐𝑥 )2
2 2
−𝑐𝑠𝑐𝑥 [𝑐𝑠𝑐𝑥 + (csc 𝑥 − cot 𝑥)]
𝑓 ′ (𝑥 ) =
(1 + 𝑐𝑠𝑐𝑥 )2
−𝑐𝑠𝑐𝑥 [𝑐𝑠𝑐𝑥 + 1]
𝑓 ′ (𝑥 ) =
(1 + 𝑐𝑠𝑐𝑥 )2
−𝑐𝑠𝑐𝑥
𝑓 ′ (𝑥 ) = 1+𝑐𝑠𝑐𝑥
 3
8. 𝑔(𝑥 ) = ln sin(4)2𝑥
1 2𝑥 3
𝑑 2𝑥3
𝑔′ (𝑥 ) = 3 ∙ cos(4) ∙ (4)
sin(4)2𝑥 𝑑𝑥

𝑑 3 3
∗∗∗ (4)2𝑥 𝑁𝑜𝑡𝑒: 𝑦 = 42𝑥
𝑑𝑥
3
ln 𝑦 = ln 42𝑥
𝑦′
= 2𝑥 3 ln 4
𝑦
3
𝑦 ′ = 6𝑥 2 ln 4 ∙ 42𝑥

2𝑥 3 2 2𝑥 3)
6 (cos 4 )(𝑥 )(ln 4)(4
𝑔′ (𝑥 ) = 3
sin(4)2𝑥

9. ℎ(𝑥 ) = 𝑒 𝑠𝑖𝑛4𝑥
′( 𝑠𝑖𝑛4𝑥
𝑑
ℎ 𝑥) = 𝑒 ∙ 𝑠𝑖𝑛4𝑥
𝑑𝑥
ℎ′ (𝑥 ) = 𝑒 𝑠𝑖𝑛4𝑥 ∙ 𝑐𝑜𝑠4𝑥 ∙ 4
ℎ′ (𝑥 ) = 4𝑒 𝑠𝑖𝑛4𝑥 cos 4𝑥
10. 𝑓(𝑥 ) = cos 2 (16 − 4𝑥 2 )
𝑓 ′ (𝑥 ) = (cos(16 − 4𝑥 2 ))2
𝑓 ′ (𝑥 ) = 2 cos(16 − 4𝑥 2 ) ∙ (− sin(16 − 4𝑥 2 ))
∙ (−8𝑥)
𝑓 ′ (𝑥 ) = 16𝑥 sin(16 − 4𝑥 2 ) cos(16 − 4𝑥 2 )
𝑓 ′ (𝑥 ) = 8𝑥(2 sin(16 − 4𝑥 2 ) cos(16 − 4𝑥 2 )
𝑓 ′ (𝑥 ) = 8𝑥𝑠𝑖𝑛2(16 − 4𝑥 2 )

11. 𝑦 = 𝑥 3 𝑐𝑜𝑠𝑥
𝑦 ′ = 3𝑥 2 𝑐𝑜𝑠𝑥 + 𝑥 3 (−𝑠𝑖𝑛𝑥 )
𝑦 ′ = 3𝑥 2 𝑐𝑜𝑠𝑥 − 𝑥 3 𝑠𝑖𝑛𝑥

𝑠𝑖𝑛𝑥
12. 𝑦= 𝑥2
2(
′
𝑥 𝑐𝑜𝑠𝑥 ) − 𝑠𝑖𝑛𝑥(2𝑥 )
𝑦 =
(𝑥 2 )2
′
𝑥(𝑥𝑐𝑜𝑠𝑥 − 2𝑠𝑖𝑛𝑥 )
𝑦 =
𝑥4
𝑥𝑐𝑜𝑠𝑥 − 2𝑠𝑖𝑛𝑥
𝑦′ =
𝑥3

1+𝑠𝑖𝑛𝑥
13. 𝑦 = 𝑥−𝑡𝑎𝑛𝑥
′
𝑥 − 𝑡𝑎𝑛𝑥(𝑐𝑜𝑠𝑥 ) − (1 + 𝑠𝑖𝑛𝑥 )(1 − sec 2 𝑥 )
𝑦 =
(𝑥 − 𝑡𝑎𝑛𝑥 )2
14. 𝑦 = sec(𝑥 2 )
𝑦 ′ = sec(𝑥 2 ) tan(𝑥 2 ) 2𝑥
𝑦 ′ = 2𝑥𝑠𝑒𝑐 (𝑥 2 )tan(𝑥 2 )

15. 𝑦 = tan(𝑠𝑖𝑛4𝑥)
𝑦 ′ = sec 2 (𝑠𝑖𝑛4𝑥 )(𝑐𝑜𝑠4𝑥 )(4)
𝑦 ′ = 4 cos(4𝑥 ) sec 2 (𝑠𝑖𝑛4𝑥)

16. y = sin2 3𝑥
𝑦 ′ = (sin(3𝑥 ))2
𝑦 ′ = 2(sin 3𝑥)(cos 3𝑥)(3)
𝑦 ′ = 6 sin(3𝑥 ) cos(3𝑥)
General Differentiation Formula
1. Derivative of a Constant

𝑑
[𝑐 ] = 0
𝑑𝑥

Examples
𝑑
[1] = 0
𝑑𝑥

𝑑
[𝜋 ] = 0
𝑑𝑥

𝑑
[−0.5] = 0
𝑑𝑥

2. Derivatives of Power Functions (Power Rule)

𝑑
If n is a positive integer, then 𝑑𝑥 [𝑥 𝑛 ] = 𝑛𝑥 𝑛−1

𝑑
a. 𝑑𝑥 [𝑥 7 ] = 7𝑥 6

𝑑
b. 𝑑𝑡 [𝑡 15 ] = 15𝑡 14
 3 1
𝑑 3
c. [𝑥 ] = 2 𝑥
2 2
𝑑𝑥

3. Derivative of a Constant Times a Function

𝑑 𝑑
[𝑐𝑓(𝑥 )] = 𝑐 ∙ [𝑓 (𝑥 )]
𝑑𝑥 𝑑𝑥

𝑑 𝑑
a. 𝑑𝑥 [5𝑥 7 ] = 5 ∙ 𝑑𝑥 [𝑥 7 ] = 5[7𝑥 6 ] = 35𝑥 6

𝑑 𝑑
b. 𝑑𝑥 [−𝑥 15 ] = (−1) ∙ 𝑑𝑥 [𝑥 15 ] = −15𝑥 14

𝑑 𝜋 𝑑 𝜋
c. [ ]=𝜋∙ [𝑥 −1 ] = 𝜋(−𝑥 −2 ) = −
𝑑𝑥 𝑥 𝑑𝑥 𝑥2

4. Derivatives of Sums or Differences

𝑑 𝑑 𝑑
(𝑓 ± 𝑔) = (𝑓) ± (𝑔)
𝑑𝑥 𝑑𝑥 𝑑𝑥

𝑑
a. 𝑑𝑥 [3𝑥 8 − 2𝑥 5 + 6𝑥 2 + 3]
𝑑 𝑑 𝑑 𝑑
[3𝑥 8 ] − [2𝑥 5 ] + [6𝑥 2 ] + [3]
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥
 = = 24𝑥 7 − 10𝑥 4 + 12𝑥

𝑑
b. 𝑑𝑥 [3𝑥 4 + 𝑥 −7 ]
𝑑 4
𝑑 −7
[3𝑥 ] + [𝑥 ]
𝑑𝑥 𝑑𝑥
= 12𝑥 3 − 7𝑥 −8
7
= 12𝑥 3 − 𝑥 8
𝑑
c. [𝑥 3 − 3𝑥 + 4]
𝑑𝑥
= 3𝑥 2 − 3

5. Derivative of a Product (Product Rule)

𝑑 𝑑𝑔 𝑑𝑓
(𝑓 ∙ 𝑔) = 𝑓 ∙ +𝑔∙
𝑑𝑥 𝑑𝑥 𝑑𝑥

1
a. 𝑓 (𝑥 ) = (3𝑥 2 + 6) (2𝑥 − 4)
𝑑 1 1 𝑑
𝑓 ′ (𝑥 ) = (3𝑥 2 + 6) ∙ (2𝑥 − ) + (2𝑥 − ) ∙ (3𝑥 2 + 6)
𝑑𝑥 4 4 𝑑𝑥
′( ) 2
1
𝑓 𝑥 = 3𝑥 + 6 2 + (2𝑥 − ) (6𝑥 )
( )( )
4
3
𝑓 ′ (𝑥 ) = 6𝑥 2 + 12 + 12𝑥 2 − 𝑥
2
 ′( 2
3
)
𝑓 𝑥 = 18𝑥 − 𝑥 + 12
2

b. 𝑓 (𝑥 ) = (2 − 𝑥 − 3𝑥 3 )(7 + 𝑥 5 )
𝑑 𝑑
𝑓 ′ (𝑥) = (2 − 𝑥 − 3𝑥 3 ) ∙ (7 + 𝑥 5 ) + (7 + 𝑥 5 ) ∙ (2 − 𝑥 − 3𝑥 3 )
𝑑𝑥 𝑑𝑥

𝑓 ′ (𝑥 ) = (2 − 𝑥 − 3𝑥 3 ) ∙ (5𝑥 4 ) + (7 + 𝑥 5 )(−1 − 9𝑥 2 )
𝑓 ′ (𝑥 ) = 10𝑥 4 − 5𝑥 5 − 15𝑥 7 − 7 − 63𝑥 2 − 𝑥 5 − 9𝑥 7
𝑓 ′ (𝑥 ) = −24𝑥 7 − 6𝑥 5 + 10𝑥 4 − 63𝑥 2 − 7

6. Derivative of a Quotient

𝑑𝑓 𝑑𝑔
𝑑 𝑓 𝑔∙ −𝑓∙
( )= 𝑑𝑥 𝑑𝑥
𝑑𝑥 𝑔 𝑔2

2𝑥−1
a. 𝑦 = 𝑥+3
𝑑 𝑑
(𝑥 + 3) ∙(2𝑥 + 1) − (2𝑥 − 1) ∙ (𝑥 + 3)
′
𝑦 = 𝑑𝑥 𝑑𝑥
(𝑥 + 3)2
′
(𝑥 + 3)(2) − (2𝑥 − 1)(1)
𝑦 =
(𝑥 + 3)2
2𝑥 + 6 − 2𝑥 + 1
𝑦′ =
(𝑥 + 3)2
 ′
7
𝑦 =
(𝑥 + 3)2

4𝑥+1
b. 𝑦 = 𝑥 2 −5
𝑑 𝑑 2
(𝑥 2 − 5) ∙ (4𝑥 + 1) − (4𝑥 + 1) ∙ (𝑥 − 5)
𝑦′ = 𝑑𝑥 𝑑𝑥
(𝑥 2 − 5)2
(𝑥 2 − 5)(4) − (4𝑥 + 1)(2𝑥)
𝑦′ =
(𝑥 2 − 5)2
4𝑥 2 − 20 − 8𝑥 2 − 2𝑥
𝑦′ =
(𝑥 2 − 5)2
−4𝑥 2 − 2𝑥 − 20
𝑦′ =
(𝑥 2 − 5)2

7. Derivative of a Power

If u is a differentiable function of x and n is any real

𝑑 𝑑𝑢
number, then 𝑑𝑥 [𝑢𝑛 ] = 𝑛 ∙ 𝑢𝑛−1 ∙ 𝑑𝑥

a. 𝑓 (𝑥 ) = [𝑥 3 ]37
𝑑 3
𝑓 ′ (𝑥 ) = 37[𝑥 3 ]36 ∙ [𝑥 ]
𝑑𝑥
𝑓 ′ (𝑥 ) = 37[𝑥 3 ]36 ∙ 3𝑥 2
𝑓 ′ (𝑥 ) = 111𝑥 110

b. 𝑓 (𝑥 ) = (3𝑥 2 + 2𝑥 )6
 𝑑
𝑓 𝑥 ) = 6(3𝑥 2 + 2𝑥 )5 ∙
′( (3𝑥 2 + 2𝑥 )
𝑑𝑥
′( ) 2 5
𝑓 𝑥 = 6(3𝑥 + 2𝑥 ) ∙ (6𝑥 + 2)
𝑓 ′ (𝑥 ) = (36𝑥 + 12)(3𝑥 2 + 2𝑥 )5

8. Derivative of a Radical with Index Equal to 2

𝑑 1 𝑑𝑢
If u is a differentiable function of x, then (√𝑢) = 2 ∙
𝑑𝑥 √ 𝑢 𝑑𝑥

a. 𝑦 = √4𝑥 8 − 1
1 𝑑
𝑦′ = ∙ (4𝑥 8 − 1)
2√4𝑥 8 − 1 𝑑𝑥
′
32𝑥 7
𝑦 =
2√4𝑥 8 − 1
′
16𝑥 7
𝑦 =
√4𝑥 8 − 1

b. 𝑦 = √5𝑥 3 + 3𝑥
′
1 𝑑
𝑦 = ∙ (5𝑥 3 + 3𝑥)
2√5𝑥 3 + 3𝑥 𝑑𝑥
′
15𝑥 2 + 3
𝑦 =
2√5𝑥 3 + 3𝑥
9. Derivative of a Radical with Index other than 2

If n is any positive integer and u is a differentiable function of x,

1 1
𝑑 1 −1 𝑑𝑢
then [𝑢 ] = 𝑛 ∙ 𝑢
𝑛 𝑛 ∙ 𝑑𝑥
𝑑𝑥

1
2
a. 𝑦 = [𝑥 − 𝑥 + 2] 3

′
1 2 −
2 𝑑
𝑦 = [𝑥 − 𝑥 + 2] 3 ∙ [𝑥 2 − 𝑥 + 2]
3 𝑑𝑥
2𝑥 − 1
𝑦′ = 2
2
3(𝑥 − 𝑥 + 2)3

1
9
b. 𝑦 = [7𝑥 + 28] 5

′
1 9 −
4 𝑑
𝑦 = [7𝑥 + 28] 5 ∙ [7𝑥 9 + 28]
5 𝑑𝑥
8
63𝑥
𝑦′ = 4
[ 9 ]
5 7𝑥 + 28 5

10. Derivative of Logarithmic Function

𝑑 1 𝑑𝑢
a. [ln 𝑥] = ∙
𝑑𝑥 𝑥 𝑑𝑥

𝑑 1 𝑑
b. [𝑙𝑜𝑔𝑏 𝑥 ] = ∙ 𝑙𝑜𝑔𝑏 𝑒 ∙ (𝑥 )
𝑑𝑥 𝑥 𝑑𝑥
Examples
a. 𝑦 = ln 5𝑥
1 𝑑
𝑦 ′ = 5𝑥 ∙ 𝑑𝑥 (5𝑥 )
5
𝑦′ =
5𝑥
1
𝑦′ =
𝑥

b. 𝑦 = 𝑥𝑙𝑛𝑥
1
𝑦 ′ = 𝑥 ∙ + ln 𝑥
𝑥
′
𝑦 = 1 + 𝑙𝑛𝑥

c. 𝑦 = 𝑥 2 𝑙𝑜𝑔2 (3 − 2𝑥 )
𝑑 𝑑 2
𝑦′ = 𝑥2 ∙ [𝑙𝑜𝑔2 (3 − 2𝑥 )] + 𝑙𝑜𝑔2 (3 − 2𝑥 ) ∙ [𝑥 ]
𝑑𝑥 𝑑𝑥
′ 2
1
𝑦 =𝑥 ∙ ∙ 𝑙𝑜𝑔2 𝑒 ∙ (−2) + 𝑙𝑜𝑔2 (3 − 2𝑥 ) ∙ (2𝑥 )
3 − 2𝑥
′
−2𝑥 2 𝑙𝑜𝑔2 𝑒
𝑦 = + 2𝑥𝑙𝑜𝑔2 (3 − 2𝑥 )
3 − 2𝑥
 11. Derivative of Exponential Function

𝑑 𝑑𝑢
a. 𝑑𝑥 [𝑎𝑢 ] = 𝑎 𝑢 ln 𝑎 ∙ 𝑑𝑥 𝑖𝑓 𝑎 𝑖𝑠 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟

𝑑 𝑑𝑢
b. 𝑑𝑥 [𝑒 𝑢 ] = 𝑒 𝑢 ∙ 𝑑𝑥

Examples:
2
a. 𝑦 = 𝑒 −5𝑥
𝑑 2
𝑦 ′ = 𝑒 −5𝑥 ∙ (−5𝑥 2 )
𝑑𝑥
2
𝑦 ′ = 𝑒 −5𝑥 ∙ (−10𝑥 )
2
𝑦 ′ = −10𝑥𝑒 −5𝑥

b. 𝑦 = 𝑒 7𝑥
′ 7𝑥
𝑑
𝑦 =𝑒 ∙ (7𝑥 )
𝑑𝑥
𝑦 ′ = 7𝑒 7𝑥

3
c. 𝑦 = 3−𝑥
3 𝑑
𝑦 ′ = 3−𝑥 ln 3 ∙ (−𝑥 3 )
𝑑𝑥
3
𝑦 ′ = −3−𝑥 ln 3 ∙ 3𝑥 2
′
3𝑥 2 ln 3
𝑦 =− 3
3𝑥
Derivatives of Inverse Trigonometric Function

𝑑 −1
1 𝑑𝑢
[sin 𝑢] = ∙
𝑑𝑥 √1 − 𝑢2 𝑑𝑥

𝑑 1 𝑑𝑢
[cos −1 𝑢] = − ∙
𝑑𝑥 √1 − 𝑢2 𝑑𝑥

𝑑 −1
1 𝑑𝑢
[tan 𝑢] = ∙
𝑑𝑥 1 + 𝑢2 𝑑𝑥

𝑑 1 𝑑𝑢
[cot −1 𝑢] = − ∙
𝑑𝑥 1 + 𝑢2 𝑑𝑥

𝑑 1 𝑑𝑢
[sec −1 𝑢] = ∙
𝑑𝑥 |𝑢|√𝑢2 − 1 𝑑𝑥

𝑑 1 𝑑𝑢
[csc −1 𝑢] = − ∙
𝑑𝑥 |𝑢|√𝑢2 − 1 𝑑𝑥
Find the derivative

1. 𝑦 = sin−1 3𝑥
1 𝑑
𝑦′ = ∙ (3𝑥 )
√1 − (3𝑥 )2 𝑑𝑥
3
𝑦′ =
√1 − 9𝑥 2

1
2. 𝑦 = sin−1 (𝑥)
1 𝑑 1
𝑦′ = ∙ ( )
𝑑𝑥 𝑥
√1 − 12
𝑥

1 1
𝑦′ = ∙ −
𝑥 2−1 𝑥2
√ 2
𝑥

′
1
𝑦 = −
𝑥√𝑥 2 − 1

3. 𝑦 = tan−1 (𝑥 3 )
1 𝑑 3
𝑦′ = ∙ (𝑥 )
1 + (𝑥 3 )2 𝑑𝑥
 2
3𝑥
𝑦′ =
1 + 𝑥6

4. 𝑦 = (𝑡𝑎𝑛𝑥 )−1
𝑑
𝑦 = −1(𝑡𝑎𝑛𝑥 )−2 ∙
′ (𝑡𝑎𝑛𝑥 )
𝑑𝑥
𝑦 = −1(𝑡𝑎𝑛𝑥 ) ∙ sec 2 𝑥
′ −2

′
sec 2 𝑥
𝑦 =−
tan2 𝑥
1
cos2 𝑥
𝑦′ = − sin2 𝑥
cos2 𝑥
1
𝑦 ′ = − sin2 𝑥

𝑦 ′ = − csc 2 𝑥

5. 𝑦 = 𝑒 𝑥 sec −1 𝑥 (𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑟𝑢𝑙𝑒)

𝑑 𝑑 𝑥
𝑦′ = 𝑒 𝑥 ∙ (sec −1 𝑥) + sec −1 𝑥 ∙ (𝑒 )
𝑑𝑥 𝑑𝑥
′
𝑒𝑥
𝑦 = + ex sec −1 𝑥
|𝑥 |√𝑥 2 − 1

6. 𝑦 = sin−1 𝑥 + cos −1 𝑥
1 𝑑 −1 𝑑
𝑦′ = ∙ ( 𝑥 ) + ∙ (𝑥 )
√1−𝑥 2 𝑑𝑥 √1−𝑥 2 𝑑𝑥
1 1
𝑦′ = −
√1 − 𝑥 2 √1 − 𝑥 2
 𝑦′ = 0

7. 𝑦 = sec −1 𝑥 + csc −1 𝑥
1 𝑑 −1 𝑑
𝑦′ = ∙ (𝑥 ) + ∙ (𝑥)
2
|𝑥|√𝑥 − 1 𝑑𝑥 2
|𝑥 |√𝑥 − 1 𝑑𝑥
1 1
𝑦′ = −
|𝑥 |√𝑥 2 − 1 |𝑥 |√𝑥 2 − 1
𝑦′ = 0

8. 𝑦 = cot −1 (√𝑥)
1 𝑑
𝑦′ = − 2 ∙ 𝑑𝑥 (√𝑥)
1−(√𝑥)

′
1 1
𝑦 = − ∙
1 − 𝑥 2 √𝑥
1
𝑦′ =
2√𝑥(1 − 𝑥 )

𝑥+1
9. 𝑦 = cos −1 ( )
2
′
1 𝑑 𝑥+1
𝑦 =− ∙ ( )
2 𝑑𝑥 2
√1 − (𝑥 + 1)
2
′
1 2
𝑦 = − ∙
2 4
√4 − (𝑥 + 1)
4
1 1
𝑦′ = − ∙
√4 − (𝑥 + 1)2 2
2
 ′
1
𝑦 = −
√4 − (𝑥 + 1)2

10. y = sec −1 (𝑥 5 )
1 𝑑 5
𝑦′ = ∙ (𝑥 )
5 5
|𝑥 |√(𝑥 ) − 12 𝑑𝑥
′
5𝑥 4
𝑦 =
|𝑥 5 |√𝑥 10 − 1

11. 𝑦 = sin−1 (2𝑥 )

′
1 𝑑
𝑦 = ∙ (2𝑥)
√1 − (2𝑥 ) 2 𝑑𝑥
′
2
𝑦 =
√1 − 4𝑥 2

12. 𝑦 = sin−1 √𝑥
′
1 𝑑
𝑦 = ∙ (√𝑥)
2 𝑑𝑥
√1 − (√𝑥)

′
1
𝑦 =
2√𝑥 − 𝑥 2

13. 𝑦 = tan−1 3𝑥
1 𝑑
𝑦′ = ∙ (3𝑥)
1 + (3𝑥 )2 𝑑𝑥
′
3
𝑦 =
1 + 9𝑥 2
14. 𝑦 = sec −1 (𝑒 2𝑥 )
1 𝑑 2𝑥
𝑦′ = ∙ (𝑒 )
2𝑥 2𝑥
|𝑒 |√(𝑒 ) − 1 2 𝑑𝑥
′
2𝑒 2𝑥
𝑦 =
𝑒 2𝑥 (√(𝑒 2𝑥 )2 − 1
2
𝑦′ =
√𝑒 4𝑥 −1

15. 𝑦 = cos −1 (2𝑥 + 1)

′
1 𝑑
𝑦 =− ∙ (2𝑥 + 1)
√1 − (2𝑥 + 1) 2 𝑑𝑥
′
2
𝑦 =−
√1 − (2𝑥 + 1)2
1. Integral of Differential of a Function
∫ 𝑑𝑥 = 𝑥 + 𝐶 𝑤ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑎𝑛 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Examples:
∫ 𝑑𝑡 = 𝑡 + 𝐶

∫ 1𝑑𝑦 = 𝑦 + 𝐶

∫ 2𝑑𝑧 = 2𝑧 + 𝐶

2. Power Rule of Integration

𝑛
𝑥 𝑛+1
∫ 𝑥 𝑑𝑥 = +𝐶 𝑤ℎ𝑒𝑟𝑒 𝑛 ≠ −1
𝑛+1

Examples:
𝑥4
a. ∫ 𝑥 3 𝑑𝑥 = +𝑐
4

1 𝑥 −5+1 1
b. ∫ 5 𝑑𝑥 = ∫ 𝑥 −5 𝑑𝑥 = +𝐶 =− +𝐶
𝑥 −5+1 4𝑥 4

1
1 +1 3 3
𝑥2 2 2
c. ∫ √𝑥 𝑑𝑥 = ∫ 𝑥 𝑑𝑥 =2 1 + 𝐶 = 𝑥 2 + 𝐶 = (√𝑥) + 𝐶
+1 3 3
2

3. Integration Using U-Substitution

Examples:
a. ∫ 2𝑥(𝑥 2 + 1)50 𝑑𝑥

𝑙𝑒𝑡 𝑢 = 𝑥 2 + 1
𝑑𝑢 = 2𝑥𝑑𝑥
 𝑑𝑢
= 𝑑𝑥
2𝑥
50
𝑢50+1 (𝑥 2 + 1)51
∫ 𝑢 𝑑𝑢 = +𝐶 = +𝐶
50 + 1 51

b. ∫ 3𝑥 2 √𝑥 3 + 3 𝑑𝑥
𝑙𝑒𝑡 𝑢 = 𝑥 3 + 3
𝑑𝑢 = 3𝑥 2 𝑑𝑥
𝑑𝑢
= 𝑑𝑥
3𝑥 2
1
1 𝑢2+1 𝑢3/2 2 3
∫ √𝑢𝑑𝑢 = ∫ 𝑢2 𝑑𝑢 = +𝐶 = + 𝐶 = (𝑥 3 + 3)2 + 𝐶
1 3 3
+1
2 2

4. Integral of Differential of a Function with a Constant Factor

∫ 𝑐𝑓(𝑥)𝑑𝑥 = 𝑐𝐹(𝑥) + 𝐶
𝑤ℎ𝑒𝑟𝑒 𝐹(𝑥) 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑡𝑖𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑓(𝑥)

Examples:
𝑥2
a. ∫ 2𝑥𝑑𝑥 = 2 ∫ 𝑥𝑑𝑥 = 2 ( ) = 𝑥 2 + 𝐶
2
𝑥4 5𝑥 4
b. ∫ 5𝑥 3 𝑑𝑥 = 5 ∫ 𝑥 3 𝑑𝑥 = 5 ( ) = +𝐶
4 4

5. Integral of a Sum/Difference
∫[𝑓(𝑥) ± 𝑔(𝑥)]𝑑𝑥 = 𝐹(𝑥) ± 𝐺(𝑥) + 𝐶

𝑤ℎ𝑒𝑟𝑒 𝐹(𝑥)𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑡𝑖𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑣𝑒 𝑜𝑓 𝑓(𝑥) 𝑎𝑛𝑑 𝐺(𝑥) 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑡𝑖𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑔(𝑥)

Examples:
2
a. ∫ 5𝑥 + 5 𝑑𝑥
3𝑥
2 1
= 5 ∫ 𝑥𝑑𝑥 + ∫ 5 𝑑𝑥
3 𝑥
 5 2
= 𝑥2 + +𝐶
2 −12𝑥 4
5 1
= 𝑥2 − 4 + 𝐶
2 6𝑥
1
−3
b. ∫ 𝑥 − 3𝑥 + 8𝑥 2 𝑑𝑥
4
1
= ∫ 𝑥 −3 𝑑𝑥 − 3 ∫ 𝑥 4 𝑑𝑥 + 8 ∫ 𝑥 2 𝑑𝑥
5
𝑥 −2 𝑥4 𝑥3
= − 3( ) + 8( )+ 𝐶
2 5 3
4
1 12 5 8 3
= − 2 − 𝑥4 + 𝑥 + 𝐶
2𝑥 5 3

6. Integration of Exponential Functions

a. ∫ 𝑒 𝑢 𝑑𝑢 = 𝑒 𝑢 + 𝑐
𝑎𝑢
b. ∫ 𝑎𝑢 𝑑𝑢 = +𝑐 𝑤ℎ𝑒𝑟𝑒 𝑎 𝑖𝑠 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟
𝑙𝑛𝑎

Examples:
a. ∫ 𝑒 −5𝑥 𝑑𝑥
𝑙𝑒𝑡 𝑢 = −5𝑥
𝑑𝑢 = −5𝑑𝑥
𝑑𝑢
= 𝑑𝑥
−5

1
− ∫ 𝑒 𝑢 𝑑𝑢
5
1
− 𝑒 −5𝑥 + 𝐶
5
2
b. ∫ 𝑥𝑒 2𝑥 𝑑𝑥
𝑙𝑒𝑡 𝑢 = 2𝑥 2
𝑑𝑢 = 4𝑥𝑑𝑥
 𝑑𝑢
= 𝑑𝑥
4𝑥

1
= ∫ 𝑒 𝑢 𝑑𝑢
4
1 2
= 𝑒 2𝑥 + 𝐶
4

3√2𝑥
c. ∫
√2𝑥
𝑢 = √4𝑥
2𝑑𝑥
𝑑𝑢 =
2√2𝑥
𝑑𝑥
𝑑𝑢 =
√2𝑥
√2𝑥𝑑𝑢 = 𝑑𝑥
𝑢
3𝑢
∫ 3 𝑑𝑢 = +𝐶
𝑙𝑛3
3√4𝑥
= +𝐶
𝑙𝑛3

7. Integrals Involving Natural Log

𝑑𝑢
∫ = ln|𝑢| + 𝑐
𝑢

Examples:
10𝑥𝑑𝑥
a. ∫
5𝑥 2

𝑙𝑒𝑡 𝑢 = 5𝑥 2
𝑑𝑢 = 10𝑥𝑑𝑥
𝑑𝑢
= 𝑑𝑥
10𝑥
𝑑𝑢
∫ = ln|𝑢| + 𝐶
𝑢
= ln|5𝑥 2 | + 𝐶
Integrals of Trigonometric Function
1. ∫ 𝑐𝑜𝑠𝑢𝑑𝑢 = 𝑠𝑖𝑛𝑢 + 𝑐
2. ∫ 𝑠𝑖𝑛𝑢𝑑𝑢 = −𝑐𝑜𝑠𝑢 + 𝑐
3. ∫ sec 2 𝑢𝑑𝑢 = 𝑡𝑎𝑛𝑢 + 𝑐
4. ∫ csc 2 𝑢𝑑𝑢 = − cot 𝑢 + 𝑐
5. ∫ 𝑠𝑒𝑐𝑢𝑡𝑎𝑛𝑢𝑑𝑢 = 𝑠𝑒𝑐𝑢 + 𝑐
6. ∫ 𝑐𝑠𝑐𝑢𝑐𝑜𝑡𝑢𝑑𝑢 = −𝑐𝑠𝑐𝑢 + 𝑐
7. ∫ 𝑡𝑎𝑛𝑢𝑑𝑢 = ln|𝑠𝑒𝑐𝑢| + 𝑐
8. ∫ 𝑐𝑜𝑡𝑢𝑑𝑢 = ln|𝑠𝑖𝑛𝑢| + 𝑐
9. ∫ 𝑠𝑒𝑐𝑢𝑑𝑢 = ln|𝑠𝑒𝑐𝑢 + 𝑡𝑎𝑛𝑢| + 𝑐
10.∫ 𝑐𝑠𝑐𝑢𝑑𝑢 = − ln|𝑐𝑠𝑐𝑢 + 𝑐𝑜𝑡𝑢| + 𝑐

Examples:
1. ∫(3𝑠𝑖𝑛𝑥 − 2 sec 2 𝑥)𝑑𝑥
3 ∫ 𝑠𝑖𝑛𝑥𝑑𝑥 − 2 ∫ sec 2 𝑥𝑑𝑥
3(−𝑐𝑜𝑠𝑥) − 2𝑡𝑎𝑛𝑥 + 𝐶
= −3𝑐𝑜𝑠𝑥 − 2𝑡𝑎𝑛𝑥 + 𝐶

2. ∫ (sec 2 𝑥 + 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥) 𝑑𝑥
∫ sec 2 𝑥𝑑𝑥 + ∫ 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥𝑑𝑥
= 𝑡𝑎𝑛𝑥 + sec 2 𝑥 + 𝐶

𝑠𝑒𝑐𝑥
3. ∫ 𝑑𝑥
𝑐𝑜𝑠𝑥
1
∫ 𝑠𝑒𝑐𝑥 ∙ 𝑑𝑥
𝑐𝑜𝑠𝑥
∫ sec 2 𝑥𝑑𝑥
= 𝑡𝑎𝑛𝑥 + 𝐶
 𝑠𝑖𝑛𝑥
4. ∫ 2 𝑑𝑥
cos 𝑥

𝑙𝑒𝑡 𝑢 = 𝑐𝑜𝑠𝑥
𝑑𝑢 = −𝑠𝑖𝑛𝑥𝑑𝑥
𝑑𝑢
= 𝑑𝑥
−𝑠𝑖𝑛𝑥

1
= −∫ 𝑑𝑢
𝑢2
= − ∫ 𝑢−2 𝑑𝑢
𝑢−1
= −( )+𝐶
−1
1
= +𝐶
𝑢
1
= +𝐶
𝑐𝑜𝑠𝑥
= 𝑠𝑒𝑐𝑥 + 𝐶

5. ∫(1 + sin2 𝑥𝑐𝑠𝑐𝑥)𝑑𝑥

sin2 𝑥
∫1+ 𝑑𝑥
𝑠𝑖𝑛𝑥
= ∫ 𝑑𝑥 + ∫ 𝑠𝑖𝑛𝑥𝑑𝑥
= 𝑥 − 𝑐𝑜𝑠𝑥 + 𝐶

6. ∫(csc 2 𝑥 − 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥)𝑑𝑥
∫ csc 2 𝑥𝑑𝑥 − ∫ 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥𝑑𝑥
= −𝑐𝑜𝑡𝑥 − 𝑠𝑒𝑐𝑥 + 𝐶

7. ∫ 𝑐𝑠𝑐𝑥(𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑡𝑥)𝑑𝑥
∫(𝑐𝑠𝑐𝑥𝑠𝑖𝑛𝑥 + 𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥) 𝑑𝑥
𝑠𝑖𝑛𝑥 1 𝑐𝑜𝑠𝑥
∫ 𝑑𝑥 + ∫ ∙ 𝑑𝑥
𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑥
 𝑐𝑜𝑠𝑥
∫ 𝑑𝑥 + ∫ 𝑑𝑥
sin2 𝑥

𝑙𝑒𝑡 𝑢 = 𝑠𝑖𝑛𝑥
𝑑𝑢 = 𝑐𝑜𝑠𝑥𝑑𝑥
𝑑𝑢
= 𝑑𝑥
𝑐𝑜𝑠𝑥

𝑢−2
∫ 𝑑𝑥 + ∫ 𝑑𝑢
2
1
= 𝑥+ +𝐶
𝑢
1
=𝑥+ +𝐶
𝑠𝑖𝑛𝑥
= 𝑥 + 𝑐𝑠𝑐𝑥 + 𝐶

𝑑𝑦
8. ∫
𝑐𝑠𝑐𝑦

∫ 𝑠𝑖𝑛𝑦𝑑𝑦
= −𝑐𝑜𝑠𝑦 + 𝐶

2
9. ∫ (𝑥 + 2 ) 𝑑𝑥
sin 𝑥

∫ 𝑥𝑑𝑥 + 2 ∫ csc 2 𝑥 𝑑𝑥
𝑥2
= + 2(−𝑐𝑜𝑡𝑥) + 𝐶
2
𝑥2
= − 2𝑐𝑜𝑡𝑥 + 𝐶
2
𝑠𝑒𝑐𝑥+𝑐𝑜𝑠𝑥
10. ∫ 𝑑𝑥
2𝑐𝑜𝑠𝑥
𝑠𝑒𝑐𝑥 𝑑𝑥
∫ 𝑑𝑥 + ∫
2𝑐𝑜𝑠𝑥 2
1 1
∫ sec 2 𝑑𝑥 + ∫ 𝑑𝑥
2 2
 1 1
= 𝑡𝑎𝑛𝑥 + 𝑥 + 𝐶
2 2

Integrals of Inverse Trigonometric Functions

𝑑𝑢 𝑢
1. ∫ 2 2 = sin−1 + 𝑐
√𝑎 −𝑢 𝑎

𝑑𝑢 1 𝑢
2. ∫ 2 2 = tan−1 + 𝑐
𝑎 +𝑢 𝑎 𝑎

𝑑𝑢 1 𝑢
3. ∫ = sec −1 + 𝑐
𝑢√𝑢2 −𝑎2 𝑎 𝑎

Examples:
1 3
a. ∫ ( − ) 𝑑𝑥
2√1−𝑥 2 1+𝑥 2
1 𝑑𝑥 𝑑𝑥
∫ − 3∫
2 √1 − 𝑥 2 1 + 𝑥2

𝑎2 = 1 𝑢2 = 𝑥 2
𝑎=1 𝑢=𝑥

1 −1
sin 𝑥 − 3 tan−1 𝑥 + 𝐶
2

4 1+𝑥+𝑥 3
b. ∫( 2 + )𝑑𝑥
𝑥√𝑥 −1 1+𝑥 2

𝑑𝑥 1𝑑𝑥 𝑥(1 + 𝑥 2 )
4∫ +∫ +∫ 𝑑𝑥
𝑥√𝑥 2 − 1 1 + 𝑥2 (1 + 𝑥 2 )

1 −1
𝑥 1 −1
𝑥 𝑥2
4 ( sec ) + tan + +𝐶
1 1 1 1 2

−1 −1
𝑥2
= 4 sec 𝑥 + tan 𝑥+ +𝐶
2
Limits of a Function

1. lim (2𝑥 + 5) = 2(3) + 5 = 11

𝑥→3

5 5 5
2. lim ( )= =
𝑥→3 𝑥+3 3+3 6

3. lim 𝑠𝑖𝑛𝑥 = sin(0) = 0

𝑥→0

𝑥 2 −16 0
4. lim =
𝑥→4 𝑥−4 0

(𝑥 + 4)(𝑥 − 4)
= (𝑥 + 4) = 8
(𝑥 − 4)

√𝑥−3 0
5. lim =
𝑥→9 𝑥−9 0

√𝑥 − 3 √𝑥 + 3 𝑥−9 1 1
∙ = = =
𝑥 − 9 √𝑥 + 3 (𝑥 − 9)(√𝑥 + 3) √𝑥 + 3 6

𝑥 2 +7 (−3)2 +7 16 16
6. lim = = =−
𝑥→−3 2𝑥−5 2(−3)−5 −11 11

𝑥−5 0
7. lim = =0
𝑥→5 𝑥 2 25
 𝑥 2 −𝑥−12 (𝑥−4)(𝑥+3)
8. lim = = (𝑥 − 4) = −7
𝑥→−3 𝑥+3 𝑥+3

𝑡 3 −𝑡 𝑡(𝑡 2 −1)
9. lim = =𝑡=1
𝑡→1 𝑡 2 −1 𝑡 2 −1

√2−𝑥−√2 √2−𝑥−√2 √2−𝑥+√2 2−𝑥−2

10. lim = ∙ =
𝑥→0 𝑥 𝑥 √2−𝑥+√2 𝑥(√2−𝑥+√2)

1 1 1
lim − =− =−
𝑥→0 √2 − 𝑥 + √2 √2 − 0 + √2 2√2