Differential Calculus Formula Sheet, Cheat Sheet of Differential Equations

Download Differential Calculus Formula Sheet and more Cheat Sheet Differential Equations in PDF only on Docsity! 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 𝑑𝑥 sin 𝑥 = cos 𝑥 𝑑 𝑑𝑥 cos 𝑥 = −sin 𝑥 𝑑 𝑑𝑥 tan 𝑥 = sec2 𝑥 𝑑 𝑑𝑥 sec 𝑥 = sec 𝑥 tan 𝑥 𝑑 𝑑𝑥 cot 𝑥 = −csc2 𝑥 𝑑 𝑑𝑥 csc 𝑥 = − csc 𝑥 cot 𝑥 Other Common Functions 𝑑 𝑑𝑥 sin−1 𝑥 = 1 √1 − 𝑥2 𝑑 𝑑𝑥 cos−1 𝑥 = − 1 √1 − 𝑥2 𝑑 𝑑𝑥 tan−1 𝑥 = 1 1 + 𝑥2 𝑑 𝑑𝑥 𝑎𝑥 = 𝑎𝑥 ln 𝑎 𝑑 𝑑𝑥 𝑒𝑥 = 𝑒𝑥 𝑑 𝑑𝑥 ln 𝑥 = 1 𝑥 , 𝑥 > 0 𝑑 𝑑𝑥 ln|𝑥| = 1 𝑥 , 𝑥 ≠ 0 𝑑 𝑑𝑥 log𝑎 𝑥 = 1 𝑥 ln 𝑎 , 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.