Formulario De Matemáticas

EXPONENTES
a ⋅a =a p a p−q =a q a p q pq (a ) =a
p q p+q

TRIGONOMETRÍA
p p

(a⋅b) =a ⋅b p p a a = p b b

p

()
p q

a = √a

q

p

CO sen θ= HIP CA cosθ= HIP sen θ CO tan θ= = cosθ CA

LIMITES

DERIVADAS
n

1 csc ϕ= senϕ 1 sec ϕ= cosϕ cos ϕ 1 ctg ϕ= = senϕ tan ϕ

1. 2. 5. 6. 7.
lim
x →c

lim b=b
x →c

3. 4.

lim x =c
x→c

n

Notación para la derivada def:
f ' ( x) o
x →c

lim x=c
x →c

lim [b f (x )]=b lim f (x )
x→c
x→c x→c

d f dx

lim [ f (x)±g ( x)]=lim f ( x)±lim g ( x)
x→c

RAÍZ CUADRADA
√ a b= √a √b a √a = b √b

HIP

CO CA

lim [ f (x) g ( x)]=lim f ( x)lim g (x )
x→c x →c x →c

√

θ
θ 0° sen θ 0 cosθ 0

π radianes=180 °

LOGARITMOS
loga N =x ⇒ a =N loga MN =loga M +loga N M loga =loga M −loga N N r logaN =r loga N log10 N=log N y loge N =ln N
x

30° 45° 1 √2 2 2 √3 √2 2 2

60 ° 90 ° 180° 270° √3 1 0 −1 2 1 0 −1 0 2

{ }

f (x) x →c = , g ( x) lim g ( x )
x →c

lim f ( x )

Reglas de Derivación: 1. ddx [c]=0 2. d [ xn ]=n xn−1 dx

lim g ( x )≠0
x →c

3. 4. 5. 6.

lim 8. 8.x → c √ f ( x)= √lim x →c
n

n

f (x)

IDENTIDADES TRIGONOMÉTRICAS
sen (−x)=−sen x cos(−x)=cos xtan (−x)=−tan x sen (x+2π)=sen x cos( x+2 π)=cos x tan (x+2π)=tan x sen (x+π)=−sen x cos( x+π)=−cos x tan (x+π)=tan x n sen (x+nπ)=(−1) sen x n cos( x+n π)=(−1) cos x tan (x+nπ)=tan x sen (n π)=0 n cos(n π)=(−1) tan (n π)=0 sen x=cos( x− π ) 2 cos x=sen( x+ π ) 2

n n 9. limc [ f (x)] ={lim f (x )} x→ x →c

Limites relacionados con la función: Valor absoluto |x|: Exponencial e :
x

lim∣x∣=∣c∣
x→c

d [ f ( x) g ( x )]= f ( x) g ' (x)+g ( x) f ' ( x) dx d f (x) g (x ) f ' (x)− f ( x) g ' (x) 7. d x g ( x) = [ g ( x)]2

d [ f ( x)+g (x)]= f ' (x)+g ' (x) dx d [ f ( x)−g (x)]= f ' (x)−g ' (x ) dx

d [c f (x )]=c f ' ( x) dx

[ ]

ALGUNOS PRODUCTOS
a⋅(c+d )=ac +ad 2 2 (a+b)⋅ (a−b)=a −b (a+b)⋅ (a+b)=(a +b )2=a 2+2a b+b2 2 2 2 (a−b)⋅ (a−b)=(a −b ) =a −2a b+b 2 (x+b)⋅ x+d )=x +(b+d)x+bd ( (a x+b)⋅(c x+d )=a c x 2+(a d +b c) x+bd (a+b)⋅ (c+d )=ac+ad+b c+bd 3 3 2 2 3 (a+b) =a +3 a b+3a b +b (a−b)3=a 3−3 a 2 b+3a b2−b 3 2 2 2 2 (a+b+c) =a +b +c +2ab+2ac+2 bc (a−b)(a 2+ab+b 2)=a 3−b 3 3 2 2 3 4 4 (a−b)(a +a b+ab +b )=a −b (a−b)(a 4+a 3 b+a2 b2+ab 3+b4)=a 5−b 5 2 2 3 3 (a+b)(a −a b+b )=a +b 3 2 2 3 4 4 (a+b)(a −a b+a b −b )=a −b 4 3 2 2 3 4 5 5 (a+b)(a −a b+a b −a b +b )=a +b 5 43 2 2 3 4 5 6 6 (a+b)(a −a b+a b −a b +a b −b )=a −b

sen x+cos x=1 2 2 1+ctg x=csc x 2 2 tan x+1=sec x 2 1 sen x= ( 1−cos2 x) 2 1 2 cos x= ( 1+cos 2 x) 2 2 1−cos2 x tan x= (1+cos 2 x) sen 2 x=2 sen xcos x 2 2 cos 2 x=cos x−sen x 2 tan x tan 2 x= 2 1−tan x

2

2

lim ∣ f ( x )∣= lim f ( x)
x →c
x →c

∣

lim e x =e lim e
x →c f ( x)

x →c c
lim f (x)
x →c

∣

Regla de laCadena:

8.

d [ f (g (x))]= f ' (g (x)) g ' (x) dx

=e

d y d y du = . dx du dx
Derivadas de funciones especiales
d dx d dx d dx d dx
u

Logaritmo ln(x):

lim ln( x)=ln(c)
x →c

siempre que c>0.
x →c x →c

lim ln( f ( x))=ln (lim f ( x))
x →c

siempre que lim f (x )>0.
Trigonométricos:

d u dx 1 d ln(u )= u u dx e =e
u u

lim sen(x )=sen (c)
x →c x →c

lim cos(x)=cos(c)
x→c x →c

d u dx d cos(u)=−sen u u dx sen(u )=cosu
u

lim sen ( f ( x))=sen( lim f ( x)) lim cos( f ( x))=cos (lim f (x))
x →c x →c

Derivadas de más de funciones :
d dx d dx d dx d dx d dx d dx d dx d dx d dx d dx a =a ln (a )

ECUACIÓN DE LA RECTA
Ecuación para calcular la pendiente de una recta: Cuando se tiene el ángulo: ( x2, y2 )
m=tan θ m= y 2− y1 x2− x1
−1

lim sec(x)=sec(c)
x →c

lim cot( x)=cot (c )

lim tan ( x)=tan(c)
x →c

lim csc(x )=csc(c)
Si lim sen(x )≠0
x→c

x →c

FORMULA GENERAL
a x +b x+c=0 entonces : 2 −b± √b −4 a c x= 2a
2

x →c

Si lim cos(x )≠0
x →c

( x1, y1 )

θ

Si f y g son funciones casi iguales:

10. lim f ( x)=lim g ( x)
x →c x →c

COSTANTES ESPECIALES
π=3.14159265359 ... e=2.21828182846... √...