Derivadas Fundamentales
Reglas
1. Constante 𝑑
𝑑𝑥 𝑐 = 0
2. Múltiplo constante: 𝑑
𝑑𝑥 𝑐𝑓(𝑥)= 𝑐𝑓′(𝑥)
3. Suma: 𝑑
𝑑𝑥 [𝑓(𝑥)± 𝑔(𝑥)] = 𝑓′(𝑥)± 𝑔′(𝑥)
4. Producto: 𝑑
𝑑𝑥 𝑓(𝑥)𝑔(𝑥)= 𝑓(𝑥)𝑔′(𝑥)+ 𝑔(𝑥)𝑓′(𝑥)
5. Cociente: 𝑑
𝑑𝑥 𝑓(𝑥)
𝑔(𝑥)=𝑔(𝑥) 𝑓′(𝑥)−𝑓(𝑥) 𝑔′(𝑥)
[ 𝑔(𝑥) ]2
6. Cadena: 𝑑
𝑑𝑥 𝑓(𝑔(𝑥))= 𝑓′(𝑔(𝑥)) 𝑔′(𝑥)
7. Potencia: 𝑑
𝑑𝑥 𝑥𝑛= 𝑛𝑥𝑛−1
8. Potencia: 𝑑
𝑑𝑥 [ 𝑔(𝑥) ]𝑛= 𝑛[ 𝑔(𝑥) ]𝑛−1𝑔′(𝑥)
Funciones trigonométricas
9. 𝑑
𝑑𝑥 sen𝑢 = cos𝑢 𝑑𝑢
𝑑𝑥 10. 𝑑
𝑑𝑥 cos 𝑢 = − sen 𝑢 𝑑𝑢
𝑑𝑥
11. 𝑑
𝑑𝑥 tan𝑢 = sec2𝑢𝑑𝑢
𝑑𝑥 12. 𝑑
𝑑𝑥 cot𝑢 = − csc2𝑢𝑑𝑢
𝑑𝑥
13. 𝑑
𝑑𝑥 sec 𝑢 = sec 𝑢 tan 𝑢 𝑑𝑢
𝑑𝑥 14. 𝑑
𝑑𝑥 csc𝑢 = − csc𝑢 cot𝑢 𝑑𝑢
𝑑𝑥
Funciones trigonométricas inversas
15. 𝑑
𝑑𝑥 sen−1 𝑢 = 1
√1−𝑢2𝑑𝑢
𝑑𝑥 16. 𝑑
𝑑𝑥 cos−1 𝑢 = − 1
√1−𝑢2𝑑𝑢
𝑑𝑥
17. 𝑑
𝑑𝑥 tan−1 𝑢 = 1
1+𝑢2𝑑𝑢
𝑑𝑥 18. 𝑑
𝑑𝑥 cot−1 𝑢 = − 1
1+𝑢2𝑑𝑢
𝑑𝑥
19. 𝑑
𝑑𝑥 sec−1 𝑢 = 1
|𝑢|√𝑢2−1𝑑𝑢
𝑑𝑥 20. 𝑑
𝑑𝑥 csc−1 𝑢 = −1
|𝑢|√𝑢2−1 𝑑𝑢
𝑑𝑥
Funciones Hiperbólicas
21. 𝑑
𝑑𝑥 senh 𝑢 = cosh𝑢 𝑑𝑢
𝑑𝑥 22. 𝑑
𝑑𝑥 cosh 𝑢 = sinh 𝑢 𝑑𝑢
𝑑𝑥
23. 𝑑
𝑑𝑥 tanh𝑢 = sech2𝑢𝑑𝑢
𝑑𝑥 24. 𝑑
𝑑𝑥 coth𝑢 = − csch2𝑢𝑑𝑢
𝑑𝑥
25. 𝑑
𝑑𝑥 sech𝑢 = − sech𝑢 tanh𝑢 𝑑𝑢
𝑑𝑥
26. 𝑑
𝑑𝑥 csch𝑢 = − csch𝑢 coth𝑢 𝑑𝑢
𝑑𝑥
Valor Absoluto 𝑑
𝑑𝑥 |𝑢|=𝑢
|𝑢|𝑑𝑢
𝑑𝑥
Funciones Hiperbólicas inversas
27. 𝑑
𝑑𝑥 senh−1 𝑢 = 1
√𝑢2+1 𝑑𝑢
𝑑𝑥
28. 𝑑
𝑑𝑥 cosh−1 𝑢 = 1
√𝑢2−1 𝑑𝑢
𝑑𝑥
29. 𝑑
𝑑𝑥 tanh−1 𝑢 = 1
1−𝑢2𝑑𝑢
𝑑𝑥
30. 𝑑
𝑑𝑥 coth−1 𝑢 = 1
1−𝑢2𝑑𝑢
𝑑𝑥
31. 𝑑
𝑑𝑥 sech−1 𝑢 = − 1
𝑢√1−𝑢2𝑑𝑢
𝑑𝑥
32. 𝑑
𝑑𝑥 csch−1 𝑢 = − 1
|𝑢|√𝑢2+1 𝑑𝑢
𝑑𝑥
Exponenciales
33. 𝑑
𝑑𝑥 exp 𝑢 = exp 𝑢 𝑑𝑢
𝑑𝑥 34. 𝑑
𝑑𝑥 𝑏𝑢= 𝑏𝑢(ln𝑏) 𝑑𝑢
𝑑𝑥
Funciones logarítmicas
35. 𝑑
𝑑𝑥 ln|𝑢|=1
𝑢𝑑𝑢
𝑑𝑥 36. 𝑑
𝑑𝑥 log𝑏𝑢 = 1
𝑢(ln𝑏)𝑑𝑢
𝑑𝑥
Leyes de los logaritmos
ln𝑥 = log𝑒𝑥 log𝑏𝑥𝑛= 𝑛 log𝑏𝑥
log𝑏𝑥𝑦 =log𝑏𝑥 + log𝑏𝑦
log𝑏𝑥
𝑦=log𝑏𝑥 − log𝑏𝑦
Funciones hiperbólicas
senh 𝑥 = 𝑒𝑥−𝑒−𝑥
2𝑐𝑜𝑠ℎ 𝑥 = 𝑒𝑥+𝑒−𝑥
2
tanh𝑥 = senh 𝑥
cosh𝑥 coth𝑥 = cosh 𝑥
senh𝑥
sech𝑥 = 1
cosh𝑥 csch𝑥 = 1
senh𝑥
coth𝑥 = 1
tanh𝑥 cosh2𝑥 − senh2𝑥 = 1
1 + tanh2𝑥 = sech2𝑥 coth2𝑥 − 1 = csch2𝑥
senh 2𝑥 = 2senh 𝑥 cosh𝑥 cosh2𝑥 = sinh2𝑥 + cosh2𝑥
senh2𝑥 = 1
2(cosh 2𝑥 − 1) cosh2𝑥 = 1
2(1 + cosh 2𝑥)
Funciones hiperbólicas
senh 𝑥 = 𝑒𝑥−𝑒−𝑥
2𝑐𝑜𝑠ℎ 𝑥 = 𝑒𝑥+𝑒−𝑥
2
tanh𝑥 = senh 𝑥
cosh𝑥 coth𝑥 = cosh 𝑥
senh𝑥
sech𝑥 = 1
cosh𝑥 csch𝑥 = 1
senh𝑥
coth𝑥 = 1
tanh𝑥 cosh2𝑥 − senh2𝑥 = 1
1 − tanh2𝑥 = sech2𝑥 coth2𝑥 − 1 = csch2𝑥
senh2𝑥 = 2 senh𝑥 cosh𝑥 cosh 2𝑥 = sinh2𝑥 + cosh2𝑥
senh2𝑥 = 1
2(cosh 2𝑥 − 1) cosh2𝑥 = 1
2(1 + cosh 2𝑥)
