Tabla De Derivadas

T TA AB BL LA AD DE ED DE ER RIIV VA AD DA AS S FUNCIÓN

FUNCIÓN DERIVADA

FUNCIÓN

FUNCIÓN DERIVADA

a

0

sen x

cos x

x

1

sen u

u' cos u

x2

2x

cos x

− senx

xm

m ⋅ x m−1

cos u

− u' senu

f ( x ) + g( x )

f ' ( x ) + g' ( x )

tgx

k.f(x)

k.f' (x)

tgu

f ( x ) ⋅ g( x )

f ' ( x ) ⋅ g( x ) + f ( x ) ⋅ g' ( x )

cot gx

f (x) g( x ) 1 f(x)

f ' ( x ) ⋅ g( x ) − f ( x ) ⋅ g' ( x ) g2 ( x ) − f ' (x) f 2 (x)

(f o g)( x ) um

ln x ln u

sec x

tg x ⋅ sec x

f ' (g(x )) ⋅ g' (x )

sec u

u' ⋅ tg u ⋅ sec u

m ⋅ um−1 ⋅ u'

cos ec x

− cot g x ⋅ cos ec x

1 x u' u

cos ec u

− u'⋅ cot g u ⋅ cos ec u

arc sen x

lga u

1 x ln a u' u ln a

ex

ex

arc cos u

eu

u' e u

arc tg x

ax

a x . ln a

arc tg u

au

a u .ln a u'

arc ctg x

uv

v.u' ⎞ ⎛ u v ⎜ v' ln u + ⎟ u ⎠ ⎝

arc ctg u

lga x =

ln x ln a

cot g u

1 = 1 + tg 2 x 2 cos x u' cos 2 u −1 = −(1 + cot g 2 x ) 2 sen x − u' = −(1 + cot g 2u) ⋅ u' 2 sen u

a,k ,m son constantes

arc sen u

arc cos x

1 1− x 2 u' 1− u2 −1 1− x 2 − u' 1− u2 1 1+ x 2 u' 1+ u2 −1 1+ x 2 − u' 1+ u2

u,v,f,g,son funciones de la variable x

__________________________________________________________________________________________ Profesor: Edis Alberto Flores

F FÓ ÓR RM MU UL LA AS SD DE ET TR RIIG GO ON NO OM ME ET TR RIIA A senα =

cat. opuesto hipotenusa 1 cos ecα = sen α

cos α =

cat. adyacente hipotenusa 1 sec α = cos α

sen 2 α + cos 2 α = 1

1 + tg 2 α = sec 2 α

sen(α + β) = senα ⋅ cos β + cos α ⋅ senβ sen(α − β ) = senα ⋅ cos β − cos α ⋅ senβ

cos(α + β ) = cos α ⋅ cos β − senα ⋅ senβ cos(α − β ) = cos α ⋅ cos β + senα ⋅ senβ tgα + tgβ tg(α + β) = 1 − tgα ⋅ tgβ

cat. opuesto sen α = cat. adyacente cos α 1 tgα = cot g α

tgα =

1 + cot g2 α = cos ec 2 α

sen 2α = 2 ⋅ sen α ⋅ cos α cos 2α = cos 2 α − sen 2 α 2tgα 1 − tg 2 α

sen

α 1 − cos α =± 2 2

cos

α 1 + cos α =± 2 2

α 1 − cos α =± 2 1 + cos α A +B A −B A +B A −B senA + senB = 2 ⋅ sen ⋅ cos cos A + cos B = 2 ⋅ cos ⋅ cos 2 2 2 2 A +B A −B A +B A −B senA − senB = 2 ⋅ cos ⋅ sen cos A − cos B = −2 ⋅ sen ⋅ sen 2 2 2 2 (R=radio de la a b c circunferencia circunscrita Teorema de los senos: senA = senB = senC = 2R al triángulo ABC)

tg2α =

tg

Teorema del coseno: a 2 = b 2 + c 2 − 2 ⋅ b ⋅ c ⋅ cos A Área de un triángulo ABC: Fórmula de Herón: (p es el semiperímetro del triángulo)

S=

1 1 b ⋅ hb = b ⋅ a ⋅ senC 2 2

S = p(p − a )(p − b )(p − c )

S=

a ⋅b ⋅c 4 ⋅R

donde

p=

a+b+c 2

F FÓ ÓR RM MU UL LA AS SD DE EL LO OG GA AR RIIT TM MO OS S loga N = b ⇔ a b = N

a>0

loga M ⋅ N = loga M + loga N

loga 1 = 0

M = loga M − loga N N log a MN = N ⋅ log a M

loga a m = m

log a M =

Si a = 10 → loga N = log N → (log aritmos decimales )

NOTA :

loga a = 1

Si a = e → loga N = ln N → (log aritmos neperianos)

loga

logb M logb a n

⎛ 1⎞ e = lim ⎜1 + ⎟ = 2'718281.... n→ +∞ ⎝ n⎠

__________________________________________________________________________________________ Profesor: Edis Alberto Flores