lim
𝑥→𝑎𝑓 (𝑥) = 𝑓 ,(𝑥)=𝑑𝑦
𝑑𝑥 ; 𝑎 = ℛ; 𝑦 = 𝑓(𝑥)
Derivadas de una función
Derivadas de una función de función
𝑦 = 𝑓(𝑢) ; 𝑢 = 𝑓(𝑥)
1. 𝑦 = 𝑐𝑡𝑒 𝑦,= 0
2. 𝑦 = 𝑥 𝑦,= 1
2a. 𝑦 = 𝑢 𝑦,= 𝑢,
3. 𝑦 = 𝑐 . 𝑥 𝑦,= 𝑐
3a. 𝑦 = 𝑐 . 𝑢 𝑦,= 𝑐 . 𝑢,
Centro de Estudiantes
Tecnológicos
A.E.T.I.
4a. 𝑦 = 𝑢 ± 𝑣 ± 𝑤. .. 𝑦,= 𝑢,±𝑣,±𝑤
,
…
5a. 𝑦 = 𝑢 . 𝑣 𝑦,= 𝑢,.𝑣 +𝑣,.𝑢
6a. 𝑦 = 𝑢
𝑣 𝑦,=𝑢,.𝑣−𝑢.𝑣,
𝑣2
7a. 𝑦 = 𝑢𝑣 𝑦,= 𝑢,.[𝑣, ln𝑢 + 𝑣 𝑢,
𝑢]
8. 𝑦 = 𝑥𝑚 𝑦,= 𝑚 . 𝑥𝑚−1
8a. 𝑦 = 𝑢𝑚 𝑦,= 𝑚 . 𝑢𝑚−1.𝑢,
9. 𝑦 = √𝑥
𝑚 𝑦,=1
𝑚 . √𝑥𝑚−1
𝑚
9a. 𝑦 = √𝑢
𝑚 𝑦,=𝑢,
𝑚 . √𝑥𝑚−1
𝑚
10. 𝑦 = √𝑥 𝑦,=1
2 . √𝑥
10a. 𝑦 = √𝑢 𝑦,=𝑢,
2 . √𝑢
11. 𝑦 = log𝑎(𝑥) 𝑦,=1
𝑥 log𝑎𝑒
11a. 𝑦 = log𝑎(𝑢) 𝑦,=𝑢,
𝑢 log𝑎𝑒
12. 𝑦 = ln(𝑥) 𝑦,=1
𝑥
12a. 𝑦 = ln(𝑢) 𝑦,=𝑢,
𝑢
13. 𝑦 = 𝑎𝑥 𝑦,= 𝑎𝑥.ln(𝑎)
13a. 𝑦 = 𝑎𝑢 𝑦,= 𝑎𝑢.𝑢,.ln(𝑎)
14. 𝑦 = 𝑒𝑥 𝑦,= 𝑒𝑥
14a. 𝑦 = 𝑒𝑢 𝑦,= 𝑒𝑢.𝑢,
15. 𝑦 = sen(𝑥) 𝑦, = cos(𝑥)
15a. 𝑦 = sen(𝑢) 𝑦,= 𝑢,.cos(𝑢)
16. 𝑦 = cos(𝑥) 𝑦,= − sen(𝑥)
16.a 𝑦 = cos(𝑢) 𝑦,= −𝑢,.sen(𝑢)
17. 𝑦 = tg(𝑥) 𝑦,=1
cos2𝑥= 1 + tg2𝑥
= sec2𝑥
17a. 𝑦 = tg(𝑢) 𝑦,=𝑢,
cos2𝑢= 𝑢,. (1 + tg2𝑢)
= 𝑢,.(sec2𝑢)
18. 𝑦 = cotg(𝑥) 𝑦,=−1
sen2𝑥= −(1 + cotg2𝑥)
= − 𝑐𝑜sec2𝑥
18a. 𝑦 = cotg(𝑢) 𝑦,=−𝑢,
sen2𝑢= −𝑢,(1 + cotg2𝑢)
= − 𝑢,.(𝑐𝑜sec2𝑢)
19. 𝑦 = sec(𝑥) 𝑦,= sec(𝑥). tg(𝑥)
19a. 𝑦 = sec(𝑢) 𝑦,= 𝑢,.sec(𝑢). tg(𝑢)
20. 𝑦 = cosec(𝑥) 𝑦,= −cosec(𝑥). cotg(𝑥)
20a. 𝑦 = cosec(𝑢) 𝑦,= −𝑢,.cosec(𝑢).cotg(𝑢)
21. 𝑦 = arcsen(𝑥) 𝑦,=1
√1− 𝑥2
21a. 𝑦 = arcsen(𝑢) 𝑦,=𝑢,
√1− 𝑢2
22. 𝑦 = arccos(𝑥) 𝑦,=−1
√1− 𝑥2
22a. 𝑦 = arccos(𝑢) 𝑦,=−𝑢,
√1− 𝑢2
23. 𝑦 = arctg(𝑥) 𝑦,=1
1+𝑥2
23a. 𝑦 = arctg(𝑢) 𝑦,=𝑢,
1+𝑢2
24. 𝑦 = arcotg(𝑥) 𝑦,=−1
1+𝑥2
24a 𝑦 = arcotg(𝑢) 𝑦,=−𝑢,
1+𝑢2
25. 𝑦 = arcsec(𝑥) 𝑦,=1
𝑥 . √𝑥2−1
25a. 𝑦 = arcsec(𝑢) 𝑦,=𝑢,
𝑢 . √𝑢2−1
26. 𝑦 = arccosec(𝑥) 𝑦,=−1
𝑥 . √𝑥2−1
26a. 𝑦 = arccosec(𝑢) 𝑦,=−𝑢,
𝑢 . √𝑢2−1
