Fórmulas de derivación 𝑎, 𝑐, 𝑛 =constantes; 𝑢, 𝑣, 𝑥 =variables o expresiones algebraicas Método de los cuatro pasos 𝑑 𝑑𝑥 𝑓(𝑥) = lim Trascendentes 𝑓(𝑥+Δ𝑥)−𝑓(𝑥) 9. Δ𝑥 Δ𝑥→0 Fórmulas básicas 1. 2. 3. 4. 5. 6. 7. 8. 10. 𝑑 𝑐=0 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 𝑑𝑥 𝑑 11. 𝑥=1 (𝑢 ± 𝑣) = 𝑑 𝑑𝑥 12. 𝑑 𝑢 ± 𝑑𝑥 𝑣 𝑑 13. (𝑐 ⋅ 𝑣) = 𝑐 𝑣 𝑑𝑥 𝑛 𝑣 =𝑛⋅𝑣 𝑛−1 (𝑢 ⋅ 𝑣) = 𝑢 𝑢 ( )= 𝑣 √𝑣 = 𝑑𝑥 𝑑 𝑑𝑥 𝑑 14. ⋅ 𝑑𝑥 𝑣 𝑣+𝑣 𝑑 𝑑𝑥 𝑢 𝑑 𝑑 𝑣 𝑢−𝑢 𝑣 𝑑𝑥 𝑑𝑥 𝑣2 𝑣 19. 𝑣 𝑑 log 𝑎 (𝑣) = 𝑑𝑥 𝑑 𝑑 𝑣 𝑑𝑥 𝑑 20. 𝑣⋅ln(𝑎) 𝑑 𝑎𝑣 = 𝑎𝑣 ⋅ ln(𝑎) ⋅ 𝑑𝑥 𝑑𝑥 𝑣 21. 𝑑 𝑒 𝑣 = 𝑒 𝑣 ⋅ 𝑑𝑥 𝑣 𝑑𝑥 𝑑 sen(𝑣) = cos(𝑣) ⋅ 𝑑𝑥 𝑑 𝑑𝑥 22. 𝑣 𝑑 𝑑 23. cos(𝑣) = − sen(𝑣) ⋅ 𝑑𝑥 𝑣 𝑑𝑥 𝑑 𝑑 15. tan(𝑣) = sec 2 (𝑣) ⋅ 𝑑𝑥 𝑣 𝑑𝑥 16. 𝑑 𝑑𝑥 24. 𝑑 𝑑 𝑑 𝑑 𝑑 tan−1 (𝑣) = 𝑑𝑥 𝑑 𝑣 𝑑𝑥 1+𝑣 2 𝑑 csc −1 (𝑣) = − 𝑑𝑥 𝑑 sec −1 (𝑣) = 𝑑𝑥 𝑑 𝑑𝑥 cot 𝑑 𝑣 𝑑𝑥 𝑣√𝑣 2−1 𝑑 𝑣 𝑑𝑥 𝑣√𝑣 2−1 −1 ( 𝑣) = − 𝑑 𝑣 𝑑𝑥 1+𝑣 2 Regla de la cadena 𝑑 sec(𝑣) = sec(𝑣) tan(𝑣) ⋅ 𝑑𝑥 𝑣 𝑑𝑥 𝑑𝑥 𝑣 𝑑𝑥 cos −1 (𝑣) = − √1−𝑣 2 𝑑𝑥 csc(𝑣) = − csc(𝑣) cot(𝑣) ⋅ 𝑑𝑥 𝑣 𝑑 𝑣 𝑑𝑥 sen−1 (𝑣) = √1−𝑣 2 𝑑𝑥 𝑑 𝑑 18. 2√𝑣 ln(𝑣) = 𝑑𝑥 𝑑𝑥 17. 𝑑 𝑣 𝑑𝑥 𝑑 𝑑 𝑑𝑦 𝑑𝑦 𝑑𝑢 = ⋅ 𝑑𝑥 𝑑𝑢 𝑑𝑥 𝑑 cot(𝑣) = − csc 2 (𝑣) ⋅ 𝑑𝑥 𝑣 Sugerencias algebraicas 𝑎𝑥 𝑏 3 𝑎 = 𝑏𝑥 √𝑥 √𝑥 = 𝑥 1 𝑥 −𝑛 = Otras cosas útiles 1 𝑥𝑛 = 𝑥𝑛 𝑚 𝑛 √𝑥 𝑚 = 𝑥 𝑛 √𝑥 = 𝑥 3 1 𝑎 𝑥 −𝑛 𝑏𝑥 𝑎 = 𝑥 −1 𝑏 Ángulos Notables Recta punto-pendiente: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) 1 𝜽 Deg 𝜽 Rad 𝐬𝐞𝐧 𝜽 𝐜𝐨𝐬 𝜽 𝐭𝐚𝐧 𝜽 𝟎° 𝟎 0 1 0 𝝅 𝟔 𝝅 𝟒 𝝅 𝟑 𝝅 𝟐 1 2 √3 3 √3 2 √3 2 √2 2 1 2 1 0 ∞ Pendiente de la Normal: 𝑚𝑁 = − 𝑚 𝑡 𝟑𝟎° 𝑚 −𝑚1 2 Ángulo entre dos rectas: 𝜃 = tan−1 (1+𝑚 1 ⋅𝑚2 Longitud de la subtangente = 1 3 √𝑥 = 𝑥 2 𝑥 2 = 𝑥 √𝑥 ) 𝟒𝟓° 𝑦1 𝑚𝑡 𝟔𝟎° Longitud de la subnormal = 𝑦1 ⋅ 𝑚𝑡 𝟗𝟎° √2 2 1 √3 Fórmulas de Integración 𝑎, 𝑏, 𝑐, 𝑛 =constantes; 𝑢, 𝑣, 𝑥 =variables o expresiones algebraicas Básicas 1. ∫ (𝑢 ± 𝑣)𝑑𝑥 = ∫ 𝑢 𝑑𝑥 ± ∫ 𝑣 𝑑𝑥 16. ∫ sec(𝑣) 𝑑𝑣 = ln|sec(𝑣) + tan(𝑣)| + 𝑐 2. ∫ 𝑎 𝑑𝑣 = 𝑎 ∫ 𝑑𝑣 17. ∫ csc(𝑣) 𝑑𝑣 = ln|csc(𝑣) − cot(𝑣)| + 𝑐 3. ∫ 𝑑𝑥 = 𝑥 + 𝑐 4. 5. ∫ 𝑣 𝑛 𝑑𝑣 = ∫ 𝑑𝑣 𝑣 𝑣 𝑛+1 𝑛+1 𝑑𝑣 𝑣 18. ∫ √𝑎2 2 = sen−1 (𝑎) + 𝑐 −𝑣 + 𝑐; 𝑛 ≠ −1 𝑑𝑣 1 𝑣 19. ∫ 𝑎2+𝑣 2 = 𝑎 tan−1 (𝑎) + 𝑐 = ln|𝑣| + 𝑐 𝑑𝑣 1 𝑣 20. ∫ = sec −1 ( ) + 𝑐 𝑎 𝑎 𝑣√𝑣 2−𝑎 2 Trascendentes 𝑑𝑣 1 𝑣+𝑎 𝑑𝑣 1 𝑣−𝑎 21. ∫ 𝑎2−𝑣 2 = 2𝑎 ln |𝑣−𝑎| + 𝑐 𝑎𝑣 6. ∫ 𝑎𝑣 𝑑𝑣 = ln 𝑎 + 𝑐 7. ∫ 𝑒 𝑣 𝑑𝑣 = 𝑒 𝑣 + 𝑐 22. ∫ 𝑣 2−𝑎2 = 2𝑎 ln |𝑣+𝑎| + 𝑐 8. ∫ sen(𝑣) 𝑑𝑣 = − cos(𝑣) + 𝑐 Integración por partes 9. ∫ cos(𝑣) 𝑑𝑣 = sen(𝑣) + 𝑐 23. ∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢 10. ∫ sec 2 (𝑣) 𝑑𝑣 = tan(𝑣) + 𝑐 Integral definida y Teorema Fundamental del Cálculo 11. ∫ csc 2 (𝑣) 𝑑𝑣 = − cot(𝑣) + 𝑐 𝑏 24. ∫𝑎 𝑓 (𝑥)𝑑𝑥 = lim ∑𝑛𝑖=1 𝑓(𝑥𝑖∗ )Δ𝑥 12. ∫ sec(𝑣) tan(𝑣) 𝑑𝑣 = sec(𝑣) + 𝑐 𝑛→∞ 𝑏−𝑎 13. ∫ csc(𝑣) cot(𝑣) 𝑑𝑣 = − csc(𝑣) + 𝑐 Δ𝑥 = 14. ∫ tan(𝑣) 𝑑𝑣 = ln|sec(𝑣)| + 𝑐 𝑥𝑖 = 𝑎 + 𝑖 ⋅ Δ𝑥 15. ∫ cot(𝑣) 𝑑𝑣 = ln|sen(𝑣)| + 𝑐 𝑛 𝑏 25. ∫𝑎 𝑓(𝑥)𝑑𝑥 = 𝐹 (𝑏) − 𝐹(𝑎 ) Sustitución trigonométrica Expresión Sustitución Identidad 𝜋 ඥ𝒂𝟐 − 𝒗𝟐 𝑣 = 𝑎 sen 𝜃; − 2 ≤ 𝜃 ≤ ඥ𝒂𝟐 + 𝒗𝟐 𝑣 = 𝑎 tan 𝜃; − 2 < 𝜃 < ඥ𝒗𝟐 − 𝒂𝟐 𝜋 𝑣 = 𝑎 sec 𝜃; 0 ≤ 𝜃 < 𝜋 2 𝜋 1 − sen2 𝜃 = cos2 𝜃 2 𝜋 1 + tan2 𝜃 = sec 2 𝜃 2 ó𝜋≤𝜃< 3𝜋 2 sec 2 𝜃 − 1 = tan2 𝜃 Notación sigma Progresiones Aritméticas Geométricas 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑟 𝑎𝑛 = 𝑎1 ⋅ 𝑟 𝑛−1 𝑆𝑛 = 𝑛(𝑎1 +𝑎𝑛) 𝑆𝑛 = 2 𝑎 𝑥ҧ = ∑𝑛𝑖=1 ( 𝑛𝑖) = 𝑎1+𝑎𝑛 2 𝑟⋅𝑎𝑛−𝑎1 ∑𝑛𝑖=1 𝑐 = 𝑛𝑐 ∑𝑛𝑖=1 𝑐𝑎𝑖 = 𝑐 ∑𝑛𝑖=1 𝑎𝑖 ∑𝑛𝑖=1(𝑎𝑖 ± 𝑏𝑖 ) = ∑𝑛𝑖=1(𝑎𝑖 ) ± ∑𝑛𝑖=1(𝑏𝑖 ) 𝑟−1 𝑥ҧ = 𝑛ඥς𝑛𝑖=1(𝑎𝑖 ) = 𝑎1 ⋅ √𝑟 𝑛−1 Interés compuesto Geométricas infinitas 𝐶 = 𝑐 (1 + 𝑟)𝑡 1 𝑆 = 1−𝑟 𝑎 ∑𝑛𝑖=1(𝑖 ) = 𝑛(𝑛+1) ∑𝑛𝑖=1(𝑖 2 ) = 2 𝑛(𝑛+1)(2𝑛+1) ∑𝑛𝑖=1(𝑖 3 ) = ቂ 6 𝑛(𝑛+1) 2 2 ቃ