ULTIMATE SUMMARY of DIFFERENTIATION & INTEGRATION all formulae and proof exam cheat sheet

Mono-differentiation/integration
a/b/n - some number
c - constant of indefinite integration
k - a constant
p - some +ve number

Proof 𝑦 𝑑𝑦 Proof
∫ 𝑦 𝑑𝑥 𝑑𝑥



Algebra
𝑛+1 𝑛 𝑛−1
𝑥
+𝑐 𝑥 𝑛𝑥
𝑛+1

𝑛+1 𝑛 𝑛−1
(𝑎𝑥+𝑏)
+𝑐 (𝑎𝑥 + 𝑏) 𝑎𝑛(𝑎𝑥 + 𝑏)
𝑎(𝑛+1)

𝑛+1 𝑛 𝑛−1
(𝑓(𝑥))
+𝑐 (𝑓(𝑥)) 𝑛(𝑓'(𝑥))(𝑓(𝑥))
(𝑓'(𝑥))(𝑛+1)

𝑛+1 𝑛 𝑛−1 2 𝑛 𝑛−1
(𝑓(𝑥)) 𝑑𝑢 𝑑𝑣
+𝑐 (𝑓'(𝑥))(𝑓(𝑥)) (𝑓(𝑥)) ((𝑓(𝑥))(𝑓''(𝑥)) + 𝑛(𝑓'(𝑥)) ) 𝑢 = 𝑓'(𝑥), 𝑑𝑥
= 𝑓''(𝑥); 𝑣 = (𝑓(𝑥)) , 𝑑𝑥
= 𝑛(𝑓'(𝑥))(𝑓(𝑥))
𝑛+1
𝑑𝑦 𝑑𝑢 𝑑𝑣 𝑛 2 𝑛−1 𝑛−1 2
𝑑𝑥
=𝑣 𝑑𝑥
+𝑢 𝑑𝑥
= (𝑓''(𝑥))(𝑓(𝑥)) + 𝑛(𝑓'(𝑥)) (𝑓(𝑥)) = (𝑓(𝑥)) ((𝑓(𝑥))(𝑓''(𝑥)) + 𝑛(𝑓'(𝑥)) )//

Exponentials & logarithms
𝑥 𝑥 𝑥 𝑥
𝑝 𝑥 𝑙𝑛 𝑝 𝑥 𝑙𝑛 𝑝 𝑑𝑦 𝑥 𝑙𝑛 𝑝 𝑥
+𝑐 𝑝 𝑝 𝑙𝑛 𝑝 𝑦 =𝑝 =𝑒 =𝑒 , = 𝑙𝑛 𝑝 𝑒 = 𝑝 𝑙𝑛 𝑝//
𝑙𝑛 𝑝 𝑑𝑥

𝑘𝑥 𝑘𝑥 𝑘𝑥
𝑝
+𝑐 𝑝 𝑘𝑝 𝑙𝑛 𝑝
𝑘 𝑙𝑛 𝑝

, 𝑘𝑥+𝑝 𝑘𝑥+𝑝 𝑘𝑥+𝑝
𝑝
+𝑐 𝑝 𝑘𝑝 𝑙𝑛 𝑝
𝑘 𝑙𝑛 𝑝

𝑓(𝑥) 𝑓(𝑥) 𝑓(𝑥)
𝑝
+𝑐 𝑝 (𝑓'(𝑥))𝑝 𝑙𝑛 𝑝
(𝑓'(𝑥))𝑙𝑛 𝑝

𝑥 𝑥 𝑥 𝑥 𝑑𝑦 𝑥 𝑑𝑦 𝑥
𝑒 +𝑐 𝑒 𝑒 𝑦 =𝑒, 𝑑𝑥
= 𝑒 𝑙𝑛 𝑒; ∵ 𝑙𝑛 𝑒 = 1, ∴ 𝑑𝑥
= 𝑒 //

𝑒
𝑘𝑥
𝑒
𝑘𝑥
𝑘𝑒
𝑘𝑥 By chain rule:
𝑘
+𝑐 𝑢 𝑑𝑢 𝑑𝑦 𝑢 𝑑𝑦 𝑢 𝑘𝑥
𝑢 = 𝑘𝑥, 𝑦 = 𝑒 , 𝑑𝑥
= 𝑘, 𝑑𝑢
=𝑒 , 𝑑𝑥
= 𝑘𝑒 = 𝑘𝑒 //

𝑒
𝑘𝑥+𝑝
𝑒
𝑘𝑥+𝑝
𝑘𝑒
𝑘𝑥+𝑝 By chain rule:
𝑘
+𝑐 𝑢 𝑑𝑢 𝑑𝑦 𝑢 𝑑𝑦 𝑢 𝑘𝑥+𝑝
𝑢 = 𝑘𝑥 + 𝑝, 𝑦 = 𝑒 , 𝑑𝑥
= 𝑘, 𝑑𝑢
=𝑒 ; 𝑑𝑥
= 𝑘𝑒 = 𝑘𝑒 //

𝑓(𝑥) 𝑓(𝑥) 𝑓(𝑥)
𝑒
+𝑐 𝑒 (𝑓'(𝑥))𝑒
𝑓'(𝑥)


𝑥 𝑙𝑛 𝑥 − 𝑥 + 𝑐 𝑙𝑛 𝑥 1 𝑑𝑦 1
𝑥
By inverse differentiation ( 𝑑𝑥 = 𝑑𝑥 ):
( 𝑑𝑦 )
𝑦 𝑑𝑥 𝑦 𝑑𝑦 1 1
If 𝑦 = 𝑙𝑛 𝑥, then 𝑥 = 𝑒 ; ∵ 𝑑𝑦
=𝑒 ,∴ 𝑑𝑥
= 𝑦 = 𝑥
//
𝑒


𝑥 𝑙𝑛 𝑘𝑥 − 𝑥 + 𝑐 𝑙𝑛 𝑘𝑥 1 𝑑
𝑥
𝑙𝑛 𝑘𝑥 = 𝑙𝑛 𝑥 + 𝑙𝑛 𝑘 (where 𝑑𝑥
𝑙𝑛 𝑘 = 0)

𝑝 𝑙𝑛|𝑘𝑥+𝑝| 𝑙𝑛(𝑘𝑥 + 𝑝) 𝑘
𝑥 𝑙𝑛(𝑘𝑥 + 𝑝) − 𝑥 + 𝑘
+𝑐 𝑘𝑥+𝑝


2 2 1
𝑙𝑛(𝑥 + 𝑥 −𝑎 ) 2
𝑥 −𝑎
2




2 2 1
𝑙𝑛(𝑥 + 𝑥 +𝑎 ) 2
𝑎 +𝑥
2

, 1 𝑎+𝑥 1
2𝑎
𝑙𝑛| 𝑎−𝑥
| 2 2
𝑎 −𝑥

1 𝑥−𝑎 1
2𝑎
𝑙𝑛| 𝑥+𝑎
| 2 2
𝑥 −𝑎

𝑛 𝑓'(𝑥)
𝑙𝑛(𝑓(𝑥)) 𝑛
(𝑓(𝑥))

𝑎 𝑎 𝑎
𝑏
𝑙𝑛|𝑥| + 𝑐 𝑏𝑥
− 2
𝑏𝑥

𝑎 𝑎 𝑎𝑏
𝑏
𝑙𝑛(|𝑏𝑥 + 𝑛|) + 𝑐 𝑏𝑥+𝑛
− 2
(𝑏𝑥+𝑛)


2 2 1
𝑙𝑛|𝑥 + 𝑥 +𝑎 | + 𝑐 2
𝑥 +𝑎
2




2 2 1
𝑙𝑛|𝑥 + 𝑥 −𝑎 | + 𝑐 2
𝑥 −𝑎
2




Trigonometry

− 𝑐𝑜𝑠 𝑥 + 𝑐 𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 𝑥 From 1st principles:
𝑑𝑦 𝑠𝑖𝑛(𝑥+ℎ)−𝑠𝑖𝑛 𝑥 𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 ℎ+𝑐𝑜𝑠 𝑥 𝑠𝑖𝑛 ℎ−𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 ℎ−1 𝑠𝑖𝑛 ℎ
𝑑𝑥
= lim ( ℎ
) = lim ( ℎ
) = lim (𝑠𝑖𝑛 𝑥 ( ℎ ) + 𝑐𝑜𝑠 𝑥 ℎ
)
ℎ→0 ℎ→0 ℎ→0
𝑐𝑜𝑠 ℎ−1 𝑠𝑖𝑛 ℎ
= 𝑠𝑖𝑛 𝑥 lim ( ℎ ) + 𝑐𝑜𝑠 𝑥 lim ( ℎ ) = 𝑐𝑜𝑠 𝑥//
ℎ→0 ℎ→0


−
𝑐𝑜𝑠 𝑘𝑥
+𝑐 𝑠𝑖𝑛 𝑘𝑥 𝑘 𝑐𝑜𝑠 𝑘𝑥 By chain rule:
𝑘 𝑑𝑢 𝑑𝑦 𝑑𝑦
𝑢 = 𝑘𝑥, 𝑦 = 𝑠𝑖𝑛 𝑢, 𝑑𝑥
= 𝑘, 𝑑𝑢
= 𝑐𝑜𝑠 𝑢; 𝑑𝑥
= 𝑘 𝑐𝑜𝑠 𝑘𝑥//

𝑐𝑜𝑠(𝑎𝑥+𝑏) 𝑠𝑖𝑛(𝑎𝑥 + 𝑏) 𝑎 𝑐𝑜𝑠(𝑎𝑥 + 𝑏)
− 𝑎
+𝑐

𝑐𝑜𝑠(𝑓(𝑥)) 𝑠𝑖𝑛(𝑓(𝑥)) (𝑓'(𝑥))𝑐𝑜𝑠(𝑓(𝑥))
− 𝑓'(𝑥)
+𝑐