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Title: Derivatives & Primitives Sheet
Description: A sheet containing all the rules for derivatives and primitives (including substitution and Leibniz rule) and 2 pages of examples.

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Derivatives
𝐷𝑒𝑓𝑖𝑛𝑖𝑛𝑔 𝑢(𝑡) 𝑎𝑛𝑑 𝑣(𝑡) 𝑎𝑠 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑡 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑟𝑢𝑙𝑒𝑠 𝑎𝑝𝑝𝑙𝑦:











𝜕
𝜕𝑡
𝜕
𝜕𝑡
𝜕
𝜕𝑡
𝜕
𝜕𝑡
𝜕
𝜕𝑡
𝜕
𝜕𝑡
𝜕
𝜕𝑡
𝜕
𝜕𝑡
𝜕
𝜕𝑡

𝑇𝑟𝑖𝑔𝑜𝑛𝑜𝑚𝑒𝑡𝑟𝑖𝑐
(𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) = 0
(𝑎𝑡 + 𝑏) = 𝑎
𝜕

(𝑢 + 𝑣 ) =
(𝑢
...
𝜕𝑡 𝑣
𝜕𝑡
𝜕

(𝑎𝑢) = 𝑎
𝑛)

𝜕

𝑢 + 𝜕𝑡 𝑣
𝜕𝑡

𝜕𝑡

𝑛

(𝑢 ∘ 𝑣 ) =

𝜕𝑡

𝜕

𝑐𝑜𝑠(𝑢) = − 𝜕𝑡 (𝑢)
...
𝜕𝑡 𝑣

𝜕

𝑎𝑢 = 𝜕𝑡 (𝑢)
...
𝑒 𝑢
𝑒
=
𝜕𝑡
𝜕𝑡



𝜕
𝜕𝑡

(𝑢

𝑣)

=

𝜕
𝑣
...
𝜕𝑡 𝑢

𝐿𝑜𝑔𝑎𝑟𝑖𝑡ℎ𝑚𝑖𝑐



𝜕
𝜕𝑡
𝜕
𝜕𝑡

ln(𝑢) =

𝜕
𝑢
𝜕𝑡
𝜕
𝑢
𝜕𝑡

log 𝑎 (𝑢) = 𝑢
...
sec(𝑢)
...
𝑐𝑜𝑡(𝑢)
𝐼𝑛𝑣𝑒𝑟𝑠𝑒 𝑇𝑟𝑖𝑔𝑜𝑛𝑜𝑚𝑒𝑡𝑟𝑖𝑐






𝑢

(u)
...
csc(𝑢)2

𝜕
𝑢
𝜕𝑡
𝑛 𝑛−1
𝑛
...
𝜕𝑡𝑣
𝜕𝑡
𝑣2

𝜕

𝑠𝑖𝑛(𝑢) = 𝜕𝑡 (𝑢)
...
𝑢𝑛−1
...
√𝑢2 −1

csc(𝑡)−1 = −

𝜕
𝑢
𝜕𝑡
𝑢
...

2
...

4
...

6
...

8
...

10
...

𝜕

(𝑡 2 + 2𝑡) = 2𝑡 + 2

𝜕𝑡
𝜕

(𝑡 2
...
𝑒 2𝑡 + 𝑡 2
...
𝑒 2𝑡 −𝑡 2
...
cos⁡(𝑡 2 )

𝜕𝑡
𝜕

𝜕𝑡

sin(𝑡 2 )4 = 4
...
2𝑡
...
𝑐𝑠𝑐 (2𝑡)
...
𝑐𝑜𝑠(𝑢) 𝑑𝑡 = 𝑠𝑖𝑛(𝑢)
𝑑𝑡

𝑑

𝑛+1
𝑢

 ∫ 𝑢
...
𝑐𝑜𝑡(𝑢) 𝑑𝑡 = − ln|𝑠𝑖𝑛(𝑢)|
𝑑𝑡

𝑑

 ∫ 𝑢
...
𝑎𝑢 𝑑𝑡 =
𝑑𝑡
ln(𝑎)

 ∫

𝑑

 ∫ 𝑢
...
𝑢𝑛 𝑑𝑡 =
𝑑𝑡

 ∫

𝑇𝑟𝑖𝑔𝑜𝑛𝑜𝑚𝑒𝑡𝑟𝑖𝑐

𝑑
𝑢
𝑑𝑡

𝑢

𝑑

 ∫ 𝑢
...
𝑐𝑠𝑐(𝑢)2 𝑑𝑡 = −𝑐𝑜𝑡(𝑢)
𝑑𝑡

𝑑𝑡 = ln |𝑢|

𝑑
𝑢
𝑑𝑡
1+𝑢2
𝑑
𝑢
𝑑𝑡

𝑑𝑡 = 𝑡𝑎𝑛

−1

𝑑

 ∫ 𝑢
...
csc(𝑢) 𝑑𝑡 = ln|𝑐𝑠𝑐 (𝑢) +
𝑑𝑡
𝑐𝑜𝑡(𝑢)|

∫ 𝑢
...
𝑢
𝑑𝑡
𝑑𝑡

𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑅𝑢𝑙𝑒: ∫ 𝑢(𝑡) 𝑑𝑡 = ∫ 𝑢(𝑣 (𝑡))
...
𝑠𝑖𝑛(𝑥)

√𝑎 2 + 𝑡 2

𝑡 = 𝑎
...
𝑠𝑒𝑐(𝑥)

MATHTOPICS | Hermano Valido

Integral Examples
1
...
∫(2𝑡)5 𝑑𝑡 = 2 ∫ 2
...
∫ 𝑒 𝑑𝑡 = 7 ∫ 7
...
∫ 3+𝑡 6 𝑑𝑡 = 3 ∫ 3+𝑡 6 𝑑𝑡 = 𝑙𝑛 |3 + 𝑡 6 | + 𝑐 , 𝑐 ∈ ℝ
2𝑡

5
...
∫ 𝑐𝑜𝑠(3𝑡) 𝑑𝑡 = 3 ∫ 3
...
∫ 𝑐𝑜𝑠(𝑡)3 𝑑𝑡 = ∫ 𝑐𝑜𝑠(𝑡)
...
(1 − 𝑠𝑖𝑛(𝑡))2 𝑑𝑡 =
= ∫ 𝑐𝑜𝑠(𝑡) − 𝑐𝑜𝑠(𝑡)
...
∫ 𝑒 2𝑡
...
𝑡 − ∫

𝑒 2𝑡
1

2

𝑑𝑡 =

9
...
𝑡𝑎𝑛(𝑥))2
1

∫𝑎
=∫

10
...
𝑡 −

...
𝑅𝑈𝐿𝐸)

𝑑𝑥 =

...
SUBS
...
𝑠𝑒𝑐 (𝑡)2 𝑑𝑡 =
= 𝑠𝑒𝑐 (𝑡)
...
(𝑠𝑒𝑐 (𝑡)2 − 1) 𝑑𝑡 =
= 𝑠𝑒𝑐 (𝑡)
...
𝑡𝑎𝑛(𝑡) + 𝑙𝑛 | 𝑠𝑒𝑐 (𝑡) + 𝑡𝑎𝑛(𝑡)| − ∫ 𝑠𝑒𝑐 (𝑡)3 𝑑𝑡

Thus ∫ 𝑠𝑒𝑐 (𝑡)3 𝑑𝑡 = 𝑠𝑒𝑐 (𝑡)
...
𝑡𝑎𝑛(𝑡) + 𝑙𝑛 | 𝑠𝑒𝑐 (𝑡) + 𝑡𝑎𝑛(𝑡)| ⟺
1

⟺ ∫ 𝑠𝑒𝑐 (𝑡)3 𝑑𝑡 = [𝑠𝑒𝑐 (𝑡)

Title: Derivatives & Primitives Sheet
Description: A sheet containing all the rules for derivatives and primitives (including substitution and Leibniz rule) and 2 pages of examples.

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