This document contains tables listing functions and their derivatives. It includes 43 entries showing common functions like polynomials, trigonometric functions, exponentials and logarithms along with their derivatives. The derivatives are expressed in terms of the original functions.

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
⋅⋅=