Derivadas
𝒚 = cte.
Integrales
[𝒇𝟏 (𝒙) ± 𝒇𝟐 (𝒙) ± ⋯ ± 𝒇𝒏 (𝒙)]dx=
𝒚′ = 0
=
𝒚 =x
𝒚′ = 1
𝒚= 𝒖
𝒚′ = 𝒖′
𝒚 = 𝒄. 𝒙
𝒚′ = c
𝒚 = 𝒄. 𝒖
𝒚′ = 𝒄. 𝒖′
𝒚= 𝒖±𝒗±𝒘±⋯
𝒚′ = 𝒖′ ± 𝒗′ ± 𝒘′ ± ⋯
𝒚 = 𝒖.v
𝒚′ = 𝒖'.v + u.v'
𝒖
y=
𝒗
𝒖´. 𝒗 −𝒖 . 𝒗´
y′ =
𝒗𝟐
𝑪𝒆𝒏𝒕𝒓𝒐 𝒅𝒆 𝑬𝒔𝒕𝒖𝒅𝒊𝒂𝒏𝒕𝒆𝒔
𝑼𝒏𝒊𝒗𝒆𝒓𝒔𝒊𝒕𝒂𝒓𝒊𝒐𝒔 𝑻𝒆𝒄𝒏𝒐𝒍𝒐𝒈𝒊𝒄𝒐𝒔
𝑰𝒏𝒅𝒆𝒑𝒆𝒏𝒅𝒊𝒆𝒏𝒕𝒆𝒔 𝑼𝑻𝑵 − 𝑭𝑹𝑽𝑴
𝒚 = 𝒙𝒎
𝒚′ = 𝒎. 𝒙𝒎−𝟏
𝟏
. 𝒍𝒐𝒈𝒂 (𝒆)
𝒙
𝟏
𝒚′ = 𝒍
𝒙
𝒚 = 𝒍𝒐𝒈𝒂 (𝒙)
𝒚′ =
𝒚 = 𝒍𝒏 (𝒙)
𝒚′ = 𝒎. 𝒖𝒎−𝟏 . 𝒖′
𝒚 = 𝒖𝒎
𝒚 = 𝒍𝒐𝒈𝒂 (𝒖)
𝒚′ =
𝒚 = 𝒍𝒏 (𝒖)
𝟏 ′
. 𝒖 . 𝒍𝒐𝒈𝒂 (𝒆)
𝒖
𝟏
𝒚′ = . 𝒖′𝒍
𝒖
𝒚 = 𝒂𝒙
𝒚′ = 𝒂𝒙 . 𝑳𝒏(𝒂)
𝒚 = 𝒂𝒖
𝒚′ = 𝒂𝒖 . 𝒖′. 𝑳𝒏(𝒂)
𝒚 = 𝒆𝒙
𝒚′ = 𝒆𝒙
𝒚 = 𝒆𝒖
𝒚′ = 𝒆𝒖 . 𝒖′
𝒚 = 𝒔𝒆𝒏 (𝒙)
𝒚′ = 𝒄𝒐𝒔 (𝒙)
𝒚 = 𝒔𝒆𝒏 (𝒖)
𝒚′ = 𝒖′ . 𝒄𝒐𝒔 (𝒖)
𝒚 = 𝒄𝒐𝒔 (𝒖)
𝒚′ = −𝒖′ . 𝒔𝒆𝒏 (𝒖)
𝒚 = 𝒄𝒐𝒔 (𝒙)
𝒚′ = −𝒔𝒆𝒏 (𝒙)
𝒚 = 𝒕𝒈 (𝒙)
𝟏
𝒚′ =
𝒄𝒐𝒔𝟐 (𝒙)
𝟐
𝟏
𝒚'= 𝟐 . 𝒖′
𝒄𝒐𝒔 (𝒖)
𝒚 = 𝒕𝒈 (𝒖)
𝟐
= (𝟏 + 𝒕𝒈 𝒙)=sec (𝒙)
𝒚′ =
𝒚 = 𝒄𝒐𝒕𝒈 (𝒙)
−𝟏
𝒔𝒆𝒏𝟐 (𝒙)
= −(𝟏 + 𝒄𝒐𝒕𝒈𝟐 𝒙)=−cosec𝟐 (𝒙)
𝒚 = 𝒔𝒆𝒄 (𝒖)
𝒚 = −𝒄𝒐𝒔𝒆𝒄 𝒙 . 𝒄𝒐𝒕𝒈(𝒙)
𝒚 = 𝒄𝒐𝒔𝒆𝒄 (𝒖)
𝒚 = 𝒂𝒓𝒄𝒔𝒆𝒏 (𝒙)
𝒚 = 𝒂𝒓𝒄𝒄𝒐𝒔 (𝒙)
𝒚 = 𝒂𝒓𝒄𝒕𝒈 (𝒙)
𝒚 = 𝒂𝒓𝒄𝒄𝒐𝒕𝒈 (𝒙)
𝒚 = 𝒂𝒓𝒄𝒔𝒆𝒄 (𝒙)
𝒚 = 𝒂𝒓𝒄𝒄𝒐𝒔𝒆𝒄 (𝒙)
′
𝒚′ =
𝒚′ =
𝟏
𝟏 − 𝒙𝟐
−𝟏
𝟏 − 𝒙𝟐
𝟏
𝒚′ =
𝟏 + 𝒙𝟐
−𝟏
𝒚′ =
𝟏 + 𝒙𝟐
𝟏
𝒚′ =
𝒙. 𝒙𝟐 − 𝟏
𝒚'=
−𝟏
𝒙. 𝒙𝟐 −𝟏
𝒚′ =
−𝟏
.u'
𝒔𝒆𝒏𝟐 (𝒖)
= −𝒖′(𝟏 + 𝒄𝒐𝒕𝒈𝟐 𝒖)=−u′ . cosec 𝟐 (𝒖)
𝒚′ = 𝒔𝒆𝒄 𝒙 . 𝒕𝒈(𝒙)
𝒚 = 𝒔𝒆𝒄 (𝒙)
𝒚 = 𝒄𝒐𝒔𝒆𝒄 (𝒙)
= 𝒖′(𝟏 + 𝒕𝒈𝟐 𝒖)= u′ . sec𝟐 (𝒖)
𝒚 = 𝒄𝒐𝒕𝒈 (𝒖)
𝒚′ = 𝒖′. 𝒔𝒆𝒄 𝒖 . 𝒕𝒈(𝒖)
′
𝒚 = −𝒖. 𝒄𝒐𝒔𝒆𝒄 𝒖 . 𝒄𝒐𝒕𝒈(𝒖)
𝒚 = 𝒂𝒓𝒄𝒔𝒆𝒏 (𝒖)
𝒚′ =
𝒚 = 𝒂𝒓𝒄𝒄𝒐𝒔 (𝒖)
′
𝟏 − 𝒖𝟐
−𝟏
. 𝒖′
𝒚 = 𝒂𝒓𝒄𝒔𝒆𝒄 (𝒖)
. 𝒖′
𝟏 − 𝒖𝟐
𝟏
𝒚′ =
. 𝒖′
𝟏 + 𝒖𝟐
−𝟏
𝒚′ =
. 𝒖′
𝟏 + 𝒖𝟐
𝟏
𝒚′ =
. 𝒖′
𝒖. 𝒖𝟐 − 𝟏
𝒚 = 𝒂𝒓𝒄𝒄𝒐𝒔𝒆𝒄 (𝒖)
𝒚'=
𝒚 = 𝒂𝒓𝒄𝒕𝒈 (𝒖)
𝒚 = 𝒂𝒓𝒄𝒄𝒐𝒕𝒈 (𝒖)
𝒚 =
𝟏
−𝟏
𝒖. 𝒖𝟐 −𝟏
𝒇𝟐 𝒙 . 𝒅𝒙 ± ⋯ ±
𝒖. 𝒗′ = 𝒖. 𝒗 −
𝟏
𝒚′ = 𝒖𝒗 [𝒗′ . 𝑳𝒏 𝒖 + 𝒗. . 𝒖′]
𝒖
𝒚 = 𝒖𝒗
(𝒇𝟏 𝒙 . 𝒅𝒙 ±
. 𝒖′
𝒗. 𝒖′
𝒅𝒙 = 𝒙 + 𝒄
𝒙𝒎 . 𝒅𝒙 =
𝒇𝒏 (𝒙).dx
(int. por partes)
(c= cte. de integracion)
𝒙𝒎+𝟏
+ 𝒄 (∀𝒎 ≠ −𝟏)
𝒎+𝟏
𝑪𝒆𝒏𝒕𝒓𝒐 𝒅𝒆 𝑬𝒔𝒕𝒖𝒅𝒊𝒂𝒏𝒕𝒆𝒔
𝑼𝒏𝒊𝒗𝒆𝒓𝒔𝒊𝒕𝒂𝒓𝒊𝒐𝒔 𝑻𝒆𝒄𝒏𝒐𝒍𝒐𝒈𝒊𝒄𝒐𝒔
𝑰𝒏𝒅𝒆𝒑𝒆𝒏𝒅𝒊𝒆𝒏𝒕𝒆𝒔 𝑼𝑻𝑵 − 𝑭𝑹𝑽𝑴
𝒖𝒎 . 𝒖′ . 𝒅𝒙 =
𝒖𝒎+𝟏
+ 𝒄 (∀𝒎 ≠ −𝟏)
𝒎+𝟏
𝟏
. 𝒅𝒙 = 𝑳𝒏(𝒙) + 𝒄
𝒙
𝟏 ′
. 𝒖 . 𝒅𝒙 = 𝑳𝒏(𝒖) + 𝒄
𝒖
𝒆𝒙 . 𝒅𝒙 = 𝒆𝒙 + 𝒄
𝒆𝒖 . 𝒖′. 𝒅𝒙 = 𝒆𝒖 + 𝒄
𝒂𝒙 . 𝒅𝒙 =
𝒂
+𝒄
𝑳𝒏(𝒂)
𝒂𝒖 . 𝒖′. 𝒅𝒙 =
𝒂𝒖
+𝒄
𝑳𝒏(𝒂)
𝒔𝒆𝒏 𝒙 . 𝒅𝒙 = − 𝒄𝒐𝒔 𝒙 + 𝒄
𝒔𝒆𝒏 𝒖 , 𝒖′. 𝒅𝒙 = − 𝒄𝒐𝒔 𝒖 + 𝒄
𝒄𝒐𝒔 𝒙 . 𝒅𝒙 = 𝒔𝒆𝒏 𝒙 + 𝒄
𝒄𝒐𝒔 𝒖 . 𝒖′ . 𝒅𝒙 = 𝒔𝒆𝒏 𝒖 + 𝒄
𝟏
. 𝒅𝒙 = 𝒕𝒈 𝒙 + 𝒄
𝒄𝒐𝒔𝟐 (𝒙)
𝟏
. 𝒅𝒙 = −𝒄𝒐𝒕𝒈 𝒙 + 𝒄
𝒔𝒆𝒏𝟐 (𝒙)
𝟏
. 𝒖′. 𝒅𝒙 = 𝒕𝒈 𝒖 + 𝒄
𝒄𝒐𝒔𝟐 (𝒖)
𝟏
. 𝒖′. 𝒅𝒙 = −𝒄𝒐𝒕𝒈 𝒖 + 𝒄
𝒔𝒆𝒏𝟐 (𝒖)
𝐬𝐞𝐜 𝒙 . 𝒕𝒈 𝒙 . 𝒅𝒙 = 𝐬𝐞𝐜 𝒙 + 𝒄
𝐬𝐞𝐜 𝒖 . 𝒕𝒈 𝒖 . 𝒖′ . 𝒅𝒙 = 𝐬𝐞𝐜 𝒖 + 𝒄
𝐜𝐨𝐬𝐞𝐜 𝒙 . 𝒄𝒐𝒕𝒈 𝒙 . 𝒅𝒙 = −𝒄𝒐 𝐬𝐞𝐜 𝒙 + 𝒄
𝐜𝐨𝐬𝐞𝐜 𝒖 . 𝒄𝒐𝒕𝒈 𝒖 . 𝒖′. 𝒅𝒙 = −𝒄𝒐𝐬𝐞𝐜 𝒙 + 𝒄
𝟏
{
{
. 𝒅𝒙 =
𝒂𝒓𝒄 𝒔𝒆𝒏 𝒙 + 𝒄
−𝒂𝒓𝒄 𝒄𝒐𝒔 𝒙 + 𝒄
𝟏 − 𝒙𝟐
𝟏
𝒂𝒓𝒄 𝒕𝒈 𝒙 + 𝒄
. 𝒅𝒙 =
𝟐
−𝒂𝒓𝒄 𝒄𝒐𝒕𝒈 𝒙 + 𝒄
𝟏+𝒙
𝟏
𝒂𝒓𝒄 𝒔𝒆𝒄 𝒙 + 𝒄
. 𝒅𝒙 =
−𝒂𝒓𝒄 𝒄𝒐𝒔𝒆𝒄 𝒙 + 𝒄
𝟐
𝒙. 𝒙 − 𝟏
{
𝟏
𝟏 − 𝒖𝟐
. 𝒖′. 𝒅𝒙 =
{
{
𝒂𝒓𝒄 𝒔𝒆𝒏 𝒖 + 𝒄
−𝒂𝒓𝒄 𝒄𝒐𝒔 𝒖 + 𝒄
𝟏
𝒂𝒓𝒄 𝒕𝒈 𝒖 + 𝒄
. 𝒖′. 𝒅𝒙 =
𝟐
−𝒂𝒓𝒄
𝒄𝒐𝒕𝒈 𝒖 + 𝒄
𝟏+𝒖
𝟏
𝒂𝒓𝒄 𝒔𝒆𝒄 𝒖 + 𝒄
. 𝒖′. 𝒅𝒙 =
−𝒂𝒓𝒄 𝒄𝒐𝒔𝒆𝒄 𝒖 + 𝒄
𝒖. 𝒖𝟐 − 𝟏
Todo el año junto a vos !!!
{
y^′=u
〖
ñ
v ) "[v′.Ln(u)+v." 1/u.u′
〗^(
"]"