Tabla de derivadas

1. Táboa de derivadas. Páx. 1
◦ Regras de derivación:
Operación Función derivada Exemplo
y = f ( x ) + g( x ) y¢ = f ¢( x ) + g¢( x ) y = x 4 + x3 ® y¢ = 4 x3 + 3x2
y = k × f ( x ) y¢ = k × f ¢ ( x ) y = 5 x3 ® y¢ = 15x 2
y = f ( x ) × g ( x ) y¢ = f ¢( x ) × g ( x ) + f ( x ) × g¢(x )
2
2 2 1 3 2
y = x ×
x
y ¢ = x × x + x × =
x
( )
( )
f x
y
g x
=
( ) ( ) ( ) ( )
¢ × - × ¢
f x g x f x g x
[ ( )]2
y
g x
¢ =
1 3 2 2
( )
3
3 2 3
3 2 6 3
y x
x
y x x x x
x x x
=
¢ × - × - - = = =
y = f [u( x )] y¢ = f ¢[u( x )] × u¢ ( x ) ( )3 2 3 2 5 y = x ® y¢ = 2 x × 3x = 6 x
◦ Derivada das funcións elementais:
Función Función derivada Exemplo
y = [u( x )]n y¢ = n × [u( x )]n-1 × u¢ ( x ) ( )2 2 2 3 y = x ® y¢ = 2x × 2 x = 4 x
n = 0 , y = 1 y¢ = 0 y = -23 ® y¢ = 0
n = 1, y = x y¢ = 1 y = -23 x ® y¢ = -23
1
2
n = , y = u( x )
( )
¢
u x
y x x y x x
¢ - - = - ® = =
¢ = 2
2 ( )
y
u x
2 2 1
2 2
2
x - x x -
x
2 2 2
1 y
n=-1 , u ( x
)
=
( )
- ¢
u x
[ ( )]2
y
u x
¢ =
( )
( ) ( ) 2 2 2 2 2
x x y y
- - - + = ® ¢ = =
1 2 3 2 3
3 3 3
x - x x - x x -
x
1
( )
= k y
n=-k , [ u ( x
)]
- ¢
k u x
x x y y
- - - + = ® ¢ = =
¢ = [ ( )]k 1
( )
y
u x +
1 3 ( 2 3 )
6 9
2 3 3 ( 2 ) 4 3 ( 2 ) 4
3
x - x x - x x -
x
y = bu( x ) y¢ = ln(b) × bu( x ) × u¢( x ) ( ) 2 2 y = 2x + x ® y¢ = ln2 × 2x + x × 2x + 1
y = bx y¢ = ln (b) × bx y = 5x ® y¢ = ln5 × 5x
y = eu( x ) y¢ = eu( x ) × u¢ ( x ) ( ) y = e3x2-5x ® y¢ =e3x2-5x × 6x - 5
y = logb [u( x )]
( )
u x
( ) ln(b)
y
u x
¢
¢ =
2
y x x y ¢ 3 x - 2
x
= - ® =
log
( 3 2
) × ( 3 2
)
ln10
x -
x
y = ln[u( x )]
( )
( )
¢
u x
y
¢ = ( ) 2
u x
y x 3 x 2
y ¢ 3 x + 4
x
= + ® =
3 2
ln 2
x +
2
x

2. Páx. 2 Táboa de derivadas.
Función Función derivada Exemplo
y = sen[u( x )] y¢ = cos[u( x )] × u¢ ( x )
( 2
)
( ) ( )
y x
y x x x x
= +
¢ = + × = +
sen 1
cos 2 1 2 2 cos 2
1
y = cos[u( x )] y¢ = -sen[u( x )] × u¢ ( x )
( )
( )
2
2
y cos
x x
1
y 2 x sen
x x
2
x
= +
æ ö
¢ = -çç + ÷÷ × +
è ø
y = tan[u( x )] y¢ = (1 + tan2 [u( x )]) × u¢( x )
( )
( ) ( ) 2
2
tan
1 1 tan
1 tan
2 2
y x
x
y x
x x
=
+
¢ = + × =
y = arcsen[u( x )]
( )
u x
1 [ ( )]2
y
u x
¢
¢ =
-
( )
2
y arcsen
x
y
1 1 1
1 2 2
x x x x
=
¢ = × =
- -
y = arccos[u( x )]
( )
- ¢
u x
1 [ ( )]2
y
u x
¢ =
-
( 2
)
( )
( )
y = arccos x -
3
x
x
- -
2 3
2 2
1 3
y
x x
¢ =
- -
y = arctan[u( x )]
( )
¢
u x
1 [ ( )]2
y
u x
¢ =
+
( 2
)
( ) ( )
= é ù ë û
¢ = × =
y arctan ln 3
x
y x
1 6 2
2 2 2 2 2
+ x x x é ë + x
ù û
1 ln 3 3 1 ln 3