descr

1. Formulario Pág.1 de 4
© Manuel Valero
TABLAS DE DERIVADAS
“DERIVADAS INMEDIATAS”
Nº FUNCIÓN PRIMITIVA FUNCIÓN DERIVADA
1. Cy = , ℜ∈C 0'=y
2. )(xfCy ⋅= , ℜ∈C )('' xfCy ⋅= , ℜ∈C
3. .......)()( ±±= xgxfy .......)(')('' ±±= xgxfy
4. )()( xgxfy ⋅= )(')()()('' xgxfxgxfy ⋅+⋅=
5. ⋅⋅⋅⋅⋅⋅⋅⋅⋅= )()()( xhxgxfy .......)()(')()()()('' +⋅⋅+⋅⋅= xhxgxfxhxgxfy
6.
)(
)(
xg
xf
y =
)(
)(')()()('
' 2
xg
xgxfxgxf
y
⋅−⋅
=
7.
)(
1
xf
y = 2
)(
)('
'
xf
xf
y
−
=
8. n
xfy )(= , ℜ∈n )(')(' 1
xfxfny n
⋅⋅= −
, ℜ∈n
9. )(xfy =
)(2
)('
'
xf
xf
y
⋅
=
10. n xfy )(= , ℜ∈n n n
xfn
xf
y
1
)(
)('
'
−
⋅
= , ℜ∈n
11. n
m
n m
xfxfy )()( == , ℜ∈nm, n mn
xfn
xfm
y
−
⋅
⋅
=
)(
)('
' , ℜ∈nm,
12. )( xf
ay = , ℜ∈a , 0>a axfay xf
ln)('' )(
⋅⋅= , ℜ∈a , 0>a
13. )(xf
ey = )('' )(
xfey xf
⋅=
14. )(ln xfy =
)(
)('
'
xf
xf
y =
15. )(log xfy a=
axf
xf
e
xf
xf
y a
ln
1
)(
)('
log
)(
)('
' ⋅=⋅=
16. )(
)( xg
xfy = )(ln)(')()(')()(' )(1)(
xfxgxfxfxfxgy xgxg
⋅⋅+⋅⋅= −
17. )(sin xfy = )(')(cos' xfxfy ⋅=
18. )(cos xfy = )(')(sin' xfxfy ⋅−=
19. )(xtgfy = )(')(sec)('))(1()('
)(cos
1
' 22
2
xfxfxfxtgxf
xf
y ⋅=⋅+=⋅=
20. )(xctgfy = )(')(cos)('))(1()('
)(sin
1
' 22
2
xfxfecxfxctgxf
xf
y ⋅−=⋅+−=⋅
−
=
21. )(sec xfy = )(')()(sec)('
)(cos
)(sin
' 2
xfxtgfxfxf
xf
xf
y ⋅⋅=⋅=
22. )(cos xecfy = )(')()(cos)('
)(sin
)(cos
' 2
2
xfxctgfxfecxf
xf
xf
y ⋅⋅−=⋅
−
=
23. )(arcsin xfy = )('
)(1
1
'
2
xf
xf
y ⋅
−
=

2. Formulario Pág.2 de 4
© Manuel Valero
TABLAS DE DERIVADAS
24. )(arccos xfy = )('
)(1
1
'
2
xf
xf
y ⋅
−
−
=
25. )(xarctgfy = )('
)(1
1
' 2
xf
xf
y ⋅
+
=
26. )(xarcctgfy = )('
)(1
1
' 2
xf
xf
y ⋅
+
−
=
27. )(sec xfarcy = )('
1)()(
1
'
2
xf
xfxf
y ⋅
−⋅
=
28. )(arccos xecfy = )('
1)()(
1
'
2
xf
xfxf
y ⋅
−⋅
−
=
29. )(xshfy = )(')(' xfxchfy ⋅=
30. )(xchfy = )(')(' xfxshfy ⋅=
31. )(xtghfy = )(')(sec)('))(1()('
)(
1
' 22
2
xfxfhxfxtghxf
xfch
y ⋅=⋅−=⋅=
32. )(xctghfy = )(')(cos)('))(1()('
)(
1
' 22
2
xfxfechxfxctghxf
xfsh
y ⋅−=⋅−=⋅
−
=
33. )(sec xhfy = )(')()(sec)('
)(
)(
' 2
xfxtghxhfxf
xfch
xshf
y ⋅⋅−=⋅
−
=
34. )(cos xechfy = )(')()(cos)('
)(
)(
' 2
2
xfxctghfxfechxf
xfsh
xchf
y ⋅⋅−=⋅
−
=
35. )(arg xshfy = )('
)(1
1
'
2
xf
xf
y ⋅
+
=
36. )(arg xchfy = )('
1)(
1
'
2
xf
xf
y ⋅
−
=
37. )(arg xtghfy = )('
)(1
1
' 2
xf
xf
y ⋅
−
=
38. )(arg xctghfy = )('
)(1
1
' 2
xf
xf
y ⋅
−
=
39. )(secarg xhfy = )('
)(1)(
1
'
2
xf
xfxf
y ⋅
−⋅
−
=
40. )(cosarg xechfy = )('
)(1)(
1
'
2
xf
xfxf
y ⋅
+⋅
−
=
41. ))(( xgfy o= )('))(('' xfxfgy ⋅=
42. ))(())(( 0
1
00
1
xffxxff −−
== oo ;
)('
1
))(()'(
0
0
1
xf
xff =−
;
))(('
1
)()'(
0
10
1
xff
xf −
−
=
43. )(),(),( xgvvfuuhy ===
dx
dv
dv
du
du
dy
dx
dy
⋅⋅=

3. Formulario Pág.3 de 4
© Manuel Valero
TABLAS DE DERIVADAS
“DERIVADAS INMEDIATAS USUALES”
Nº FUNCIÓN PRIMITIVA FUNCIÓN DERIVADA
1. Cy = , ℜ∈C 0'=y
2. xCy ⋅= , ℜ∈C Cy =' , ℜ∈C
3. .......±±= gfy , )(,....., xFgf ∈ .......''' ±±= gfy
4. )()( xgxfy ⋅= , )(, xFgf ∈ ''' gfgfy ⋅+⋅=
5. ⋅⋅⋅⋅⋅⋅⋅⋅⋅= )()()( xhxgxfy , )(,...,, xFhgf ∈ .......''' +⋅⋅+⋅⋅= hgfhgfy
6.
g
f
y = , )(, xFgf ∈ 2
''
'
g
gfgf
y
⋅−⋅
=
7.
f
y
1
= , )(xFf ∈ 2
'
'
f
f
y
−
=
8. n
xy = , ℜ∈n 1
' −
⋅= n
xny , ℜ∈n
9. xy =
x
y
⋅
=
2
1
'
10. n
xy = , ℜ∈n n n
xn
y
1
1
'
−
⋅
= , ℜ∈n
11. n
m
n m
xxy == , ℜ∈nm, n mn
xn
m
y
−
⋅
=' , ℜ∈nm,
12. x
ay = , ℜ∈a , 0>a aay x
ln' ⋅= , ℜ∈a , 0>a
13. x
ey = x
ey ='
14. xy ln=
x
y
1
'=
15. xy alog=
ax
e
x
y a
ln
11
log
1
' ⋅=⋅=
16. g
fy = , )(, xFgf ∈ fgfffgy gg
ln''' 1
⋅⋅+⋅⋅= −
17. xy sin= xy cos'=
18. xy cos= xy sin' −=
19. tgxy = xxtg
x
y 22
2
sec1
cos
1
' =+==
20. ctgxy = xecxctg
x
y 22
2
cos)1(
sin
1
' −=+−=
−
=
21. xy sec= tgxx
x
x
y ⋅== sec
cos
sin
' 2
22. ecxy cos= ctgxxec
x
x
y ⋅−=
−
= 2
2
cos
sin
cos
'
23. xy arcsin=
2
1
1
'
x
y
−
=
24. xy arccos=
2
1
1
'
x
y
−
−
=

4. Formulario Pág.4 de 4
© Manuel Valero
TABLAS DE DERIVADAS
25. arctgxy =
2
1
1
'
x
y
+
=
26. arcctgxy =
2
1
1
'
x
y
+
−
=
27. xarcy sec=
1
1
'
2
−⋅
=
xx
y
28. ecxy arccos=
1
1
'
2
−⋅
−
=
xx
y
29. shxy = chxy ='
30. chxy = shxy ='
31. tghxy = xhxtgh
xch
y 22
2
sec1
1
' =−==
32. ctghxy = xechxctgh
xsh
y 22
2
cos1
1
' −=−=
−
=
33. hxy sec= tghxhx
xch
shx
y ⋅−=
−
= sec' 2
34. echxy cos= ctghxxech
xsh
chx
y ⋅−=
−
= 2
2
cos'
35. shxy arg=
2
1
1
'
x
y
+
=
36. chxy arg=
1
1
'
2
−
=
x
y
37. tghxy arg=
2
1
1
'
x
y
−
=
38. ctghxy arg=
2
1
1
'
x
y
−
=
39. hxy secarg=
2
1
1
'
xx
y
−⋅
−
=
40. echxy cosarg=
2
1
1
'
xx
y
+⋅
−
=
41. ))(( xgfy o= )('))(('' xfxfgy ⋅=
42. ))(())(( 0
1
00
1
xffxxff −−
== oo ;
)('
1
))(()'(
0
0
1
xf
xff =−
;
))(('
1
)()'(
0
10
1
xff
xf −
−
=
43. )(),(),( xgvvfuuhy ===
dx
dv
dv
du
du
dy
dx
dy
⋅⋅=