Differential Calculus Formula Sheet
DERIVATIVES
𝐼𝑓 𝑦 = 𝑓(𝑥),𝑡ℎ𝑒𝑛 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑
𝑓′(𝑥)= lim
𝑛→∞
𝑓(𝑥 + ℎ)− 𝑓(𝑥)
ℎ
𝑓′(𝑥)= 𝑦′=𝑑𝑓
𝑑𝑥 =𝑑𝑦
𝑑𝑥 =𝑑
𝑑𝑥(𝑓(𝑥))=𝐷𝑓(𝑥)
Basic Properties
1. 𝑑
𝑑𝑥 (𝑐)= 0
2. 𝑑
𝑑𝑥 (𝑥𝑛)= 𝑛𝑥𝑛−1 Power Rule
3. (𝑐𝑓(𝑥))′= 𝑐𝑓′(𝑥)
4. (𝑓(𝑥)± 𝑔(𝑥))′= 𝑓′(𝑥) ± 𝑔′(𝑥)
5. (𝑓(𝑥)𝑔(𝑥))′= 𝑓′(𝑥)𝑔(𝑥)+ 𝑓(𝑥)𝑔′(𝑥)
Product Rule
6. (𝑓(𝑥)
𝑔(𝑥))′=𝑓′(𝑥)𝑔(𝑥)−𝑓(𝑥)𝑔′(𝑥)
(𝑔(𝑥)2
Quotient Rule
7. 𝑑
𝑑𝑥 (𝑓(𝑔(𝑥)) = 𝑓′(𝑔(𝑥))𝑔′(𝑥)
Chain Rule
8. 𝑑
𝑑𝑥 (𝑥)= 1
Derivatives of Common
Trigonometric Functions
𝑑
𝑑𝑥sin𝑥 = cos 𝑥
𝑑
𝑑𝑥cos𝑥 = −sin 𝑥
𝑑
𝑑𝑥tan𝑥 = sec2𝑥
𝑑
𝑑𝑥sec𝑥 = sec 𝑥tan 𝑥
𝑑
𝑑𝑥cot𝑥 = −csc2𝑥
𝑑
𝑑𝑥csc𝑥 = − csc𝑥 cot𝑥
Other Common Functions
𝑑
𝑑𝑥sin−1 𝑥 = 1
√1 − 𝑥2
𝑑
𝑑𝑥cos−1 𝑥 = − 1
√1 − 𝑥2
𝑑
𝑑𝑥tan−1 𝑥 = 1
1 + 𝑥2
𝑑
𝑑𝑥𝑎𝑥= 𝑎𝑥ln 𝑎
𝑑
𝑑𝑥𝑒𝑥= 𝑒𝑥
𝑑
𝑑𝑥ln𝑥 = 1
𝑥,𝑥 > 0
𝑑
𝑑𝑥ln|𝑥|=1
𝑥,𝑥 ≠ 0
𝑑
𝑑𝑥log𝑎𝑥 = 1
𝑥ln𝑎, 𝑥 > 0
Technique
𝐿𝑒𝑡 𝑓(𝑥)= 𝑢
𝐿𝑒𝑡 𝑔(𝑥)= 𝑣
𝑢 ∗ 𝑣 = 𝑢′𝑣 + 𝑢𝑣′
* The Arrangement of U And V Does
Not Matter.
Technique
𝐿𝑒𝑡 𝑓(𝑥)= 𝑢
𝐿𝑒𝑡 𝑔(𝑥)= 𝑣
𝑢
𝑣=𝑢′𝑣 − 𝑢𝑣′
𝑣2
* The Arrangement of U And V Matter.
