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Formulario Pág. 1 de 4
“DERIVADAS INMEDIATAS”
Nº FUNCIÓN PRIMITIVA FUNCIÓN DERIVADA
1. y = C , C ∈ℜ y '= 0 2. y = C ⋅ f ( x ), C ∈ℜ y ' = C ⋅ f '( x ), C ∈ℜ 3. y^ =^ f (^ x )± g ( x )±....... y '^ =^ f '( x )± g '( x )±....... 4. y^ =^ f (^ x )⋅ g ( x ) y^ '^ = f '( x )⋅ g ( x )+ f ( x )⋅ g '( x ) 5. y^ =^ f (^ x )⋅ g ( x )⋅ h ( x )⋅⋅⋅⋅⋅⋅⋅ y '^ =^ f '( x )⋅ g ( x )⋅ h ( x )+ f ( x )⋅ g '( x )⋅ h ( x )+.......
g x
f x y = ( )
2 g x
f x g x f x g x y
f x
y = 2 ( )
f x
f x y
n y = f ( x ) , n ∈ℜ ' ( ) '( )
1 y n f x f x
n = ⋅ ⋅
− , n ∈ℜ
9. y = f ( x ) 2 ( )
f x
f x y ⋅
10. y = n^ f ( x ), n ∈ℜ n n n f x
f x y 1 ( )
− ⋅
= , n ∈ℜ
11. n
m n m^ y = f ( x ) = f ( x ) , m , n ∈ℜ n n m n f x
m f x y − ⋅
' , m , n ∈ℜ
f ( x ) y = a , a ∈ℜ, a > 0 y a f x a
f x ' '( ) ln
( ) = ⋅ ⋅ , a ∈ℜ, a > 0
f ( x ) y = e ' '( )
( ) y e f x
f x = ⋅
14. y^ =^ ln^ f ( x ) ( )
f x
f x y =
15. y = log (^) af ( x ) f x a
f x e f x
f x y (^) a ln
log ( )
() ( )
gx y = f x ' ( ) ( ) '( ) ( ) '( ) ln ( )
( ) 1 () y g x f x f x f x g x f x
g x gx = ⋅ ⋅ + ⋅ ⋅
−
17. y^ =^ sin^ f ( x ) y^ '^ =cos f ( x )⋅ f '( x ) 18. y = cos f ( x ) y ' =−sin f ( x )⋅ f '( x ) 19. y^ =^ tgf (^ x ) '( ) (^1 ( )) '( ) sec ( ) '( ) cos ( )
2 2 2
f x tg x f x f x f x f x
y = ⋅ = + ⋅ = ⋅
20. y^ =^ ctgf (^ x ) '( ) (^1 ( )) '( ) cos ( ) '( ) sin ( )
2 2 2
f x ctg x f x ec f x f x f x
y ⋅ =− + ⋅ =− ⋅
21. y^ =^ sec^ f ( x ) '( ) sec ( ) ( ) '( ) cos ( )
sin ( ) ' 2
f x f x tgf x f x f x
f x y = ⋅ = ⋅ ⋅
22. y^ =^ cos^ ecf ( x ) '( ) cos ( ) ( ) '( ) sin ( )
cos ( ) '
2 2 f x ec f x ctgf x f x f x
f x y ⋅ =− ⋅ ⋅
23. y^ =^ arcsin^ f ( x ) '( ) 1 ( )
2
f x f x
y ⋅ −
Formulario Pág. 2 de 4
24. y^ =^ arccos^ f ( x ) '( ) 1 ( )
2
f x f x
y ⋅ −
25. y^ =^ arctgf (^ x ) '( ) 1 ( )
2
f x f x
y ⋅
26. y^ =^ arcctgf ( x ) '( ) 1 ( )
2
f x f x
y ⋅
27. y^ =^ arc sec^ f ( x ) '( ) ( ) ( ) 1
2
f x f x f x
y ⋅ ⋅ −
28. y^ =^ arccos^ ecf ( x ) '( ) ( ) ( ) 1
2
f x f x f x
y ⋅ ⋅ −
29. y = shf ( x ) y ' = chf ( x )⋅ f '( x ) 30. y^ =^ chf (^ x ) y^ '^ = shf ( x )⋅ f '( x ) 31. y^ =^ tghf (^ x ) '( ) (^1 ( )) '( ) sec ( ) '( ) ( )
2 2 2
f x tgh x f x h f x f x ch f x
y = ⋅ = − ⋅ = ⋅
32. y^ =^ ctghf (^ x ) '( ) (^1 ( )) '( ) cos ( ) '( ) ( )
2 2 2
f x ctgh x f x ech f x f x sh f x
y ⋅ = − ⋅ =− ⋅
33. y^ =^ sec^ hf ( x ) '( ) sec ( ) ( ) '( ) ( )
2 f x hf x tghx f x ch f x
shf x y ⋅ =− ⋅ ⋅
34. y^ =^ cos^ echf ( x ) '( ) cos ( ) ( ) '( ) ( )
2 2 f x ech f x ctghf x f x sh f x
chf x y ⋅ =− ⋅ ⋅
35. y^ =^ arg^ shf ( x ) '( ) 1 ( )
2
f x f x
y ⋅
36. y^ =^ arg^ chf ( x ) '( ) ( ) 1
2
f x f x
y ⋅ −
37. y^ =^ arg^ tghf ( x ) '( ) 1 ( )
2
f x f x
y ⋅ −
38. y^ =^ arg^ ctghf ( x ) '( ) 1 ( )
2
f x f x
y ⋅ −
39. y^ =^ arg^ sec hf ( x ) '( ) ( ) 1 ( )
2
f x f x f x
y ⋅ ⋅ −
40. y^ =^ arg^ cos echf ( x ) '( ) ( ) 1 ( )
2
f x f x f x
y ⋅ ⋅ +
41. y = ( f o g )( x ) y ' = g '( f ( x ))⋅ f '( x )
42. (^ )( ) ( )( 0 )
1 0 0
1 f f x x f f x
− − o = = o ; '( )
0
0
1
f x
f f x =
− ; '( ( ))
0
(^01)
1
f f x
f x −
−
43. y^ =^ h (^ u ), u = f ( v ), v = g ( x ) dx
dv
dv
du
du
dy
dx
dy = ⋅ ⋅
Formulario Pág. 4 de 4
25. y^ =^ arctgx 2 1
x
y
26. y^ =^ arcctgx 2 1
x
y
27. y^ = arc sec x
1
2 ⋅ −
x x
y
28. y^ =arccos ecx
1
2 ⋅ −
x x
y
29. y^ =^ shx y^ '= chx 30. y^ =^ chx y^ '= shx 31. y^ =^ tghx tghx h x
ch x
y
2 2 2
1 sec
32. y^ =^ ctghx ctgh x echx sh x
y
2 2 2 1 cos
33. y^ =^ sec hx hx tghx ch x
shx y =− ⋅
' = sec 2
34. y^ =^ cos echx echx ctghx
shx
chx y =− ⋅
2 2 ' cos
35. y^ =^ arg shx 2 1
x
y
36. y^ =arg chx
1
2 −
x
y
37. y^ =^ arg tghx 2 1
x
y −
38. y^ =^ arg ctghx 2 1
x
y −
39. y^ =^ argsec hx 2 1
x x
y ⋅ −
40. y^ =^ argcos echx 2 1
x x
y ⋅ +
41. y^ =^ (^ f o g )( x ) y^ '^ = g '( f ( x ))⋅ f '( x )
1 0 0
1 f f x x f f x
− − o = = o ; '( )
0
0
1
f x
f f x =
− ; '( ( ))
0
(^01)
1
f f x
f x −
−
43. y^ =^ h (^ u ), u = f ( v ), v = g ( x )
dx
dv
dv
du
du
dy
dx
dy = ⋅ ⋅
