FORMULARIO_CALCULO DIFERENCIAL

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CÁLCULO DIFERENCIAL 
 
Valor absoluto 
𝑎 ≤ |𝑎|; −𝑎 ≤ |𝑎|; |𝑎| ≥ 0; |𝑎| = 0 
|𝑎| = {
𝑎 𝑠𝑖 𝑎 ≥ 0
−𝑎 𝑠𝑖 𝑎 < 0
 
|𝑎| = |−𝑎| 
|𝑎𝑏| = |𝑎||𝑏| 
|𝑎 + 𝑏| ≤ |𝑎| + |𝑏| 
Logaritmos 
log𝑎 𝑁 = 𝑥 → 𝑎
𝑥 = 𝑁 
log𝑎 𝑀𝑁 = log𝑎 𝑀 + log𝑎 𝑁 
log𝑎
𝑀
𝑁
= log𝑎 𝑀 − log𝑎 𝑁 
log𝑎 𝑁
𝑟 = 𝑟 log𝑎 𝑁 
log𝑎 𝑁 =
log𝑏 𝑁
log𝑏 𝑎
=
ln 𝑁
ln 𝑎
 
log10 𝑁 = log 𝑁 → log𝑒 𝑁 = ln 𝑁 
Trigonometría 
sin 𝜃 =
𝐶𝑂
𝐻𝐼𝑃
; csc 𝜃 =
1
sin 𝜃
 
cos 𝜃 =
𝐶𝐴
𝐻𝐼𝑃
; sec 𝜃 =
1
cos 𝜃
 
tan 𝜃 =
sin 𝜃
cos 𝜃
=
𝐶𝑂
𝐶𝐴
; cot 𝜃 =
1
tan 𝜃
 
𝑠𝑖𝑛2𝜃 + 𝑐𝑜𝑠2𝜃 = 1 
1 + 𝑐𝑜𝑡2𝜃 = 𝑐𝑠𝑐2𝜃 
𝑡𝑎𝑛2𝜃 + 1 = 𝑠𝑒𝑐2𝜃 
𝑠𝑖𝑛2𝜃 =
1 − cos 2𝜃
2
 
𝑐𝑜𝑠2𝜃 =
1 + cos 2𝜃
2
 
𝑡𝑎𝑛2𝜃 =
1 − cos 2𝜃
1 + cos 2𝜃
 
Funciones hiperbólicas 
sinh 𝑥 =
𝑒𝑥 − 𝑒−𝑥
2
 
cosh 𝑥 =
𝑒𝑥 + 𝑒−𝑥
2
 
tanh 𝑥 =
sinh 𝑥
cosh 𝑥
=
𝑒𝑥 − 𝑒−𝑥
𝑒𝑥 + 𝑒−𝑥
 
coth 𝑥 =
1
tanh 𝑥
=
𝑒𝑥 + 𝑒−𝑥
𝑒𝑥 − 𝑒−𝑥
 
sech 𝑥 =
1
cosh 𝑥
=
2
𝑒𝑥 + 𝑒−𝑥
 
csch 𝑥 =
1
sinh 𝑥
=
2
𝑒𝑥 − 𝑒−𝑥
 
sinh ∶ ℝ → ℝ 𝑐𝑠𝑐ℎ ∶ ℝ − {0} → ℝ − {0} 
cosh : ℝ → [1, ∞⟩ 𝑠𝑒𝑐ℎ ∶ ℝ → ⟨0,1] 
𝑡𝑎𝑛ℎ ∶ ℝ → 〈−1,1〉 
𝑐𝑜𝑡ℎ ∶ ℝ − {0} → ℝ − {0} 
Funciones hiperbólicas inversas 
sinh−1 𝑥 = ln (𝑥 + √𝑥2 + 1) , ∀𝑥𝜖ℝ 
cosh−1 𝑥 = ln (𝑥 ± √𝑥2 − 1) , 𝑥 ≥ 1 
tanh−1 𝑥 =
1
2
ln (
1 + 𝑥
1 − 𝑥
) , |𝑥| < 1 
coth−1 𝑥 =
1
2
ln (
𝑥 + 1
𝑥 − 1
) , |𝑥| > 1 
sech−1 𝑥 = ln (
1 ± √1 − 𝑥2
𝑥
), 0 < 𝑥 ≤ 1 
csch−1 𝑥 = ln (
1
𝑥
+
√𝑥2 + 1
|𝑥|
), 𝑥 ≠ 0 
Límites 
lim
𝑥→𝑎
𝐾𝑓(𝑥) = 𝐾 lim
𝑥→𝑎
𝑓(𝑥) 
lim
𝑥→𝑎
[𝑓(𝑥) ± 𝑔(𝑥)] = lim
𝑥→𝑎
𝑓(𝑥) ± lim
𝑥→𝑎
𝑔(𝑥) 
lim
𝑥→𝑎
[𝑓(𝑥) ∙ 𝑔(𝑥)] = lim
𝑥→𝑎
𝑓(𝑥) ∙ lim
𝑥→𝑎
𝑔(𝑥) 
lim
𝑥→𝑎
[
𝑓(𝑥)
𝑔(𝑥)
] =
lim
𝑥→𝑎
𝑓(𝑥)
lim
𝑥→𝑎
𝑔(𝑥)
 
lim
𝑥→𝑎
𝑥𝑛 − 𝑎𝑛
𝑥 − 𝑎
= 𝑛𝑎𝑛−1 
lim
𝑥→0
(1 + 𝑥)
1
𝑥 = 𝑒 
lim
𝑥→0
sin 𝑥
𝑥
= 1 
lim
𝑥→0
tan 𝑥
𝑥
= 1 
lim
𝑥→0
1 − cos 𝑥
𝑥
= 0 
lim
𝑥→0
𝑒𝑥 − 1
𝑥
= 1 
lim
𝑥→1
𝑥 − 1
ln 𝑥
= 1 
lim
𝑥→∞
(1 +
1
𝑥
)
𝑥
= 𝑒 
Regla de L'Hôpital 
lim
𝑥→𝑎
𝑓(𝑥)
𝑔(𝑥)
= lim
𝑥→𝑎
𝑓′(𝑥)
𝑔′(𝑥)
 
Derivadas 
𝐷𝑥𝑓(𝑥) =
𝑑𝑓
𝑑𝑥
= lim
∆𝑥→0
𝑓(𝑥 + ∆𝑥) − 𝑓(𝑥)
∆𝑥
 
𝑑
𝑑𝑥
(𝑐) = 0; 
𝑑
𝑑𝑥
(𝑐𝑥) = 𝑐 
𝑑
𝑑𝑥
(𝑐𝑥𝑛) = 𝑛𝑐𝑥𝑛−1 
𝑑
𝑑𝑥
(𝑢 ± 𝑣 ± 𝑤 ± ⋯ ) =
𝑑𝑢
𝑑𝑥
±
𝑑𝑣
𝑑𝑥
±
𝑑𝑤
𝑑𝑥
±. .. 
𝑑
𝑑𝑥
(𝑐𝑢) = 𝑐
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(𝑢𝑣) = 𝑢
𝑑𝑣
𝑑𝑥
+ 𝑣
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(𝑢𝑣𝑤) = 𝑢𝑣
𝑑𝑤
𝑑𝑥
+ 𝑢𝑤
𝑑𝑣
𝑑𝑥
+ 𝑣𝑤
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(
𝑢
𝑣
) =
𝑣(𝑑𝑢/𝑑𝑥) − 𝑢(𝑑𝑣/𝑑𝑥)
𝑣2
 
𝑑
𝑑𝑥
(𝑢𝑛) = 𝑛𝑢𝑛−1 
𝑑𝑢
𝑑𝑥
 
𝑑𝐹
𝑑𝑥
=
𝑑𝐹
𝑑𝑢
∙
𝑑𝑢
𝑑𝑥
 
𝑑𝑢
𝑑𝑥
=
1
𝑑𝑥/𝑑𝑢
; 
𝑑𝐹
𝑑𝑥
=
𝑑𝑓/𝑑𝑢
𝑑𝑥/𝑑𝑢
 
𝑑𝑦
𝑑𝑥
= 
𝑑𝑦/𝑑𝑡
𝑑𝑥/𝑑𝑡
=
𝑓′
2
(𝑡)
𝑓′
1
(𝑡)
, {
𝑥 = 𝑓1(𝑡)
𝑦 = 𝑓2(𝑡)
 
Derivadas de funciones log y exp. 
𝑑
𝑑𝑥
(ln 𝑢) =
𝑑𝑢
𝑑𝑥
𝑢
=
1
𝑢
∙
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(log 𝑢) =
log 𝑒
𝑢
∙
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(𝑙𝑜𝑔𝑎 𝑢) =
𝑙𝑜𝑔𝑎 𝑒
𝑢
∙
𝑑𝑢
𝑑𝑥
 𝑎 > 0, 𝑎 ≠ 1 
𝑑
𝑑𝑥
(𝑒𝑢) = 𝑒𝑢
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(𝑎𝑢) = 𝑎𝑢 ln 𝑎
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(𝑢𝑣) = 𝑣𝑢𝑣−1
𝑑𝑢
𝑑𝑥
+ ln 𝑢 ∙ 𝑢𝑣
𝑑𝑣
𝑑𝑥
 
Derivadas de funciones trigonométricas 
𝑑
𝑑𝑥
(sin 𝑢) = cos 𝑢
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(cos 𝑢) = −sen 𝑢
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(tan 𝑢) = 𝑠𝑒𝑐2 𝑢
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(cot 𝑢) = −𝑐𝑠𝑐2 𝑢
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(sec 𝑢) = sec(𝑢) tan (𝑢)
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
(csc 𝑢) = −csc(𝑢) tan (𝑢)
𝑑𝑢
𝑑𝑥
 
Derivadas de funciones trigonométricas 
inversas 
𝑑
𝑑𝑥
sin−1 𝑢 =
1
√1 − 𝑢2
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
cos−1 𝑢 = −
1
√1 − 𝑢2
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
tan−1 𝑢 =
1
1 + 𝑢2
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
cot−1 𝑢 = −
1
1 + 𝑢2
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
sec−1 𝑢 = ±
1
𝑢√𝑢2 − 1
𝑑𝑢
𝑑𝑥
 
𝑑
𝑑𝑥
csc−1 𝑢 = ∓
1
𝑢√𝑢2 − 1
𝑑𝑢
𝑑𝑥
 
 
{
+ 𝑠𝑖 𝑢 > 1
− 𝑠𝑖 𝑢 < −1
 
{
− 𝑠𝑖 𝑢 > 1
+ 𝑠𝑖 𝑢 < −1