TRIGONOMETRÍA
sin(𝛼 + 𝛽) = sin 𝛼 ∙ cos 𝛽 + cos 𝛼 ∙ sin 𝛽
cos(𝛼 + 𝛽) = cos 𝛼 ∙ cos 𝛽 − sin 𝛼 ∙ sin 𝛽
sin2 𝑥 + cos 2 𝑥 = 1
tan2 𝑥 + 1 = sec 2 𝑥
cosh2 𝑥 − sinh2 𝑥 = 1
sin(2𝑥) = 2 sin 𝑥 ∙ cos 𝑥
sin 𝑥
cos 𝑥
cos 𝑥
cot 𝑥 =
sin 𝑥
1
sec 𝑥 =
cos 𝑥
1
csc 𝑥 =
sin 𝑥
sin2 𝑥 =
tan 𝑥 =
1 + cos(2𝑥)
2
𝑥
𝑒 − 𝑒 −𝑥
sinh 𝑥 =
2
cos 2 𝑥 =
cosh 𝑥 =
INTEGRALES
DERIVADAS
𝑓(𝑥) = 𝑘
⟹
𝑓 ′ (𝑥) = 0
𝑓(𝑥) = 𝑥
⟹
𝑓 ′ (𝑥) = 1
∫ 𝑑𝑥 = ∫ 1 𝑑𝑥 = 𝑥 + 𝐶
𝑓(𝑥) = 𝑐𝑥
⟹
𝑓 ′ (𝑥) = 𝑐
𝑓(𝑥) = 𝑥 𝑛
⟹
𝑓 ′ (𝑥) = 𝑛 ∙ 𝑥 𝑛−1
𝑓(𝑥) = ln 𝑥
(regla del producto)
∫
1
𝑓 ′ (𝑥) =
𝑥
⟹
𝑓(𝑥) = sin 𝑥
⟹
𝑓 ′ (𝑥) = cos 𝑥
𝑓(𝑥) = cos 𝑥
⟹
𝑓 ′ (𝑥) = − sin 𝑥
𝑓(𝑥) = 𝑒 𝑥
⟹
𝑓 ′ (𝑥) =
1
𝑥 ∙ ln 𝛼
𝑓(𝑥) = tan 𝑥
⟹
𝑓 ′ (𝑥) = sec 2 𝑥
𝑓(𝑥) = cot 𝑥
⟹
𝑓 ′ (𝑥) = − csc 2 𝑥
𝑓(𝑥) = sec 𝑥
⟹
𝑓 ′ (𝑥) = sec 𝑥 ∙ tan 𝑥
𝑓(𝑥) = csc 𝑥
⟹
𝑓 ′ (𝑥) = − csc 𝑥 ∙ cot 𝑥
𝑓(𝑥) = 𝛼 𝑥
⟹
𝑓(𝑥) = arctan 𝑥
𝑓(𝑥) = arcsin 𝑥
𝑓(𝑥) = arccos 𝑥
𝑥 𝑛+1
+𝐶
𝑛+1
1
𝑑𝑥 = ln|𝑥 | + 𝐶
𝑥
∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶
∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶
𝑓 ′ (𝑥) = 𝑒 𝑥
⟹
𝑓(𝑥) = log 𝛼 𝑥
∫ 𝑥 𝑛 𝑑𝑥 =
𝑓 ′ (𝑥) = 𝛼 𝑥 ∙ ln 𝛼
𝑥
⟹ 𝑓 ′ (𝑥) =
1 + 𝑥2
1
⟹ 𝑓 ′ (𝑥) =
√1 − 𝑥 2
1
⟹ 𝑓 ′ (𝑥) = −
√1 − 𝑥 2
𝑓(𝑥) = sinh 𝑥
⟹
𝑓 ′ (𝑥) = cosh 𝑥
𝑓(𝑥) = cosh 𝑥
⟹
𝑓 ′ (𝑥) = sinh 𝑥
∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶
∫ 𝛼 𝑥 𝑑𝑥 =
𝛼𝑥
+𝐶
ln 𝛼
∫ sinh 𝑥 𝑑𝑥 = cosh 𝑥 + 𝐶
∫ cosh 𝑥 𝑑𝑥 = sinh 𝑥 + 𝐶
∫ sec 2 𝑥 𝑑𝑥 = tan 𝑥 + 𝐶
∫ csc 2 𝑥 𝑑𝑥 = − cot 𝑥 + 𝐶
∫ sec 𝑥 ∙ tan 𝑥 𝑑𝑥 = sec 𝑥 + 𝐶
∫ csc 𝑥 ∙ cot 𝑥 𝑑𝑥 = − csc 𝑥 + 𝐶
∫
∫
1
𝑑𝑥 = arctan 𝑥 + 𝐶
1 + 𝑥2
1
𝑑𝑥 = arcsin 𝑥 + 𝐶
√1 − 𝑥 2
1
∫
𝑑𝑥 = 𝐴𝑟𝑔𝑡𝑎𝑛ℎ 𝑥 + 𝐶
1 − 𝑥2
∫
∫
1
√1 + 𝑥 2
1
√𝑥 2 − 1
1 − cos(2𝑥)
2
𝑑𝑥 = 𝐴𝑟𝑔𝑠𝑖𝑛ℎ 𝑥 + 𝐶
𝑑𝑥 = 𝐴𝑟𝑔𝑐𝑜𝑠ℎ 𝑥 + 𝐶
𝑒 𝑥 + 𝑒 −𝑥
2
LOGARITMOS
𝛼 log𝛼 𝑥 = 𝑥
log 𝛼 1 = 0
log 𝑏𝑛 (𝑎𝑛 ) = log 𝑏 𝑎
log 𝛼 𝛼 = 1
log 𝑏 (𝑎𝑛 ) = 𝑛 ⋅ log 𝑏 𝑎
log 𝛼 (𝑥 ⁄𝑦) = log 𝛼 𝑥 − log 𝛼 𝑦
log 𝛼 (𝑥 ∙ 𝑦) = log 𝛼 𝑥 + log 𝛼 𝑦
log 𝑏 𝑎 =
log 𝑏 𝑛 = 𝑥 ⟺ 𝑏 𝑥 = 𝑛
1
log 𝑎 𝑏
log 𝑏 𝑎 =
log 𝑐 𝑎
log 𝑐 𝑏
VALOR ABSOLUTO
|𝑎| = |−𝑎|
|𝑎|
𝑎
| |=
|𝑏|
𝑏
|𝑎𝑏| = |𝑎||𝑏|
|𝑥 + 𝑦| ≤ |𝑥 | + |𝑦|
LÍMITES
sin 𝑥
=1
𝑥→0 𝑥
lim
lim
1 𝑥
𝑥
tan 𝑥
=1
𝑥→0 𝑥
=1
lim
𝑘 𝑥
𝑥
lim (1 + ) = 𝑒 𝑘
lim (1 + ) = 𝑒
𝑥→∞
𝑥
𝑥→0 sin 𝑥
𝑥→∞
lim (1 +
𝑥→∞
𝑘 𝑥+𝑎
)
= 𝑒𝑘
𝑥+𝑎
REGLAS DE DERIVACIÓN
𝑦 = 𝑓(𝑥) ∙ 𝑔(𝑥)
𝑓(𝑥)
𝑦=
𝑔(𝑥)
⟹
ℎ(𝑥) = 𝑓(𝑔(𝑥))
Regla de la cadena:
𝑦 ′ = 𝑓 ′ (𝑥) ∙ 𝑔(𝑥) + 𝑓(𝑥) ∙ 𝑔′ (𝑥)
⟹
𝑦′
𝑓 ′ (𝑥) ∙ 𝑔(𝑥) − 𝑓(𝑥) ∙ 𝑔′ (𝑥)
=
𝑔2 (𝑥)
⟹
ℎ′ (𝑥) = 𝑓 ′ (𝑔(𝑥)) 𝑔′(𝑥)
PROPIEDAD FUNDAMENTAL DE LA DIVISIÓN
𝐷(𝑥) = 𝑑(𝑥) ∙ 𝑐(𝑥) + 𝑅(𝑥)
𝐷(𝑥)
𝑅(𝑥)
= 𝑐(𝑥) +
𝑑(𝑥)
𝑑(𝑥)
ÁLGEBRA LINEAL
Teorema de Laplace:
Donde 𝑀𝑖,𝑗 es el determinante de la submatriz
obtenida al remover la 𝑖– é𝑠𝑖𝑚𝑎 fila y la
𝑗– é𝑠𝑖𝑚𝑎 columna de 𝐵 .
𝑛
det(𝐵) = ∑(−1)𝑖+𝑗 ⋅ 𝐵𝑖,𝑗 ⋅ 𝑀𝑖,𝑗
𝑗=1
CÓNICAS
Para saber el centro (ℎ, 𝑘) sustituir 𝑥 con (𝑥 − ℎ) e 𝑦 con (𝑦 − 𝑘)
CIRCUNFERENCIA
ELIPSE
𝑥 2 + 𝑦 2 = 𝑟2
𝑥2 𝑦2
+
=1
𝑎2 𝑏 2
HIPÉRBOLA
𝑥2 𝑦2
−
=1
𝑎2 𝑏 2
𝑦2 𝑥2
−
=1
𝑎2 𝑏 2