Tablas

Función
n
1 y=a

Derivadas y = f '(x) = dy/dx

Integrales / f(x) dx = F(x) + c

y' = 0

a.x + c

2

y =x

y'=1

x 2 / 2 +c

3

y = a.x

y' = a

a.x 2 / 2 + c

4

y = xn

y1 = n.x"-1

x n+1 /(n+1) + c

5

y = X~n

y' = _ n x -^ 1 =-n / x (n+1)

(x- n + 1 )/ (-n+1) + c

6

y = x/J =Vx

y' = 1 / (2.x7*) = 1 / (2. Vx)

(2/3)(x3/2) + c =(2/3)>/x 3 +c7

y = X a/b = b Vx a

y' = ( a/b).x (a/b) - 1

x (a/b)

8

y = 1/x

y1 = -1 / x2

Inx + c

9

y = sen x

y' = eos x

- eos x + c

10

y = c osx

y' = - sen x

sen x + c

11

y = t gx

y' = 1 / cos2x = sec2 x

- In eos x + c

12

y = cotg x

y1 = -1 / sen2x = - cosec2 x

In sen x + c

13

y = sec x

y' = sen x / cos2x = sec x . tg x

In (tg1Ax) + c

14

y = cosec x

y' = - eos x / sen2x = - cosec x. cotg x

In [eos x / (1 - sen x)] + c

15

y = are sen x

y^l/O-x 2 )' 7 ^ 1 / V O - x 2 )

x.arc sen x + (1 - X2)'7' + c

16

y = are eos x

y' = -1 / (1 - x2)* = - 1 / V (1 - x2)

x.arc eos x - (1 - x2)'72 + c

17

y = are tg x

y1 = 1 / (1 + x2)

x.arc tg x - %.ln (1 + x2) + c

18

y = are cotg xy' = -1 / (1 + x2)

x.arc cotg x + 1Aln (1 + x2) + c

19

y = are sec x

y' = 1 l[x.(y?-VVr]

x. are sec x - a. In (x+íx2-'!)1'2) + c

20

y = are cosec x

y' = -17 [x.(x2 - 1)*J

x. are cosec x - a. In (x+(x2-1)1/2) + c

21

y = senh x

y' = cosh x

cosh x + c

22

y = cosh x

y' = senh x

senh x + c

23

y - tgh x

y' = sech2x = (sechx)2

In cosh x + c24

y = cotgh x

y1 = - cosech2 x = - (cosech x)2

In senh x + c

25

y = sech x

y' = - sech x . tgh x

2. are tg ex + c

26

y = cosech x

y1 = - cosech x. cotgh x

In tgh (x/2) + c

27

y = lnx

y1 = 1 / x

x.(ln x- 1) + c

28

y = logaX

y1 = (1 / x) . In a

x.(logaX-1/lna) + c

29

y = e'

y' = ex

30

y = a'

y' = ax.ln a

ex + c
ax / Ina + c

31

y = x*

y' = x*. ( lnx + 1)

32

y = u.v

y1 = u'.v + v'.u

33

y = u/ v

y1 = (u'.v- v'.u)/ v2

34

y = uv

y' = uv. (v'.ln u + v.u' /u)

35

y = lnuv

y' = (v'.u.lnu - u'.v.lnv) / v.u.ln2 u

Integral por partes í u. dv = u.v - J v. du
Linealidad í[a.f(x) ± b.g(x )]

d x-a J f (x) dx ± b j g(x) dx

Barrow f f(x) dx = F(b) - F(a)

+ 1 /[(a/b)+1] +C

cié

en ^

Longitud arco de curva

volumen

'^^
HHI

Tabla de Derivadas e Integrales
a, b: parámetros
c: constante
u, v: f(x), funciones
n: nro real
e = 2.7182 TT = 3.1415

ÜSucST^

revolución

'

sólido

.¿^ f1'
K
J.> '\ )
L

( 0 36(s) >0 / x e E' (a, 8) => f(x) e E(l, s)

Lim
x-»0

sen x = 1
x

Lim(l+x)lx = e

Jf' (n\m - l llll
II —
ft-0

r(a +f i ) - / ( a )
h

/ U) - / (a)
i—a
i-a

X-> oo

Newton-Raphson

Si f (a)= lim x >af(x) es c ontinua en x=a sino: si 3 lim
finito discontinua evitable sino discontinuidad esencial
Derivada

L im [ l + _ L ] x = e

X-^0

tg x = 1
x

r

Jn+l

r

-T»

Integral impropia

fír \
r,/

X

\

*«

/ t*(í-ór= lin: f
/

*— -«- '« '

f(i\dji

Regla de L'Hospital:lim f(x) = f '(x) . . . .
lim O/O, oo / oo
g(x) g '(x)

Regla de la cadena:Si y = f (u) A u = g(x) => y ' = f (u). g'(x) Definición de logaritmo: l og a b = c < = > a c = b
f-b
/•!,•( S.
, Cambio de extremos de Cambio de base log a b = log b = In b
•'o
•'fia)
integración u = g (x)
log a In a
Recta tg = f '(a)-(x - a) + f(a )
Recta normal = - (x - a) / f '(a)+ f(a) Recta secante = (Ay / Ax ) • (x - a) + f (a)
jet OC —

1
sen a

LUbCt IX —

tan

eos a
sen 2 a + cos2a = 1 *n(2 a) = 2 sena cosa.
1 + tg 2 a = sec 2 a Cí w (2 a) = eos2 a - sen2 a
1 + ctg 2 a = cosec 2 tg
a
l-# 2 a

71

71

0 = 0 ° - = 3 0° _=45° _ =60° -=90°
4
3
2
6
sen a
0
1/2
V2 / 2
V3 /2
1
V2 / 2
1/2
0
eos a
1
V3 12
1
A/3
oo
tga
0
V 3/3
i

t"O" 1*1 ^^ I f i

—C...