Funciones elementales Funciones compuestas - pinae.es · PDF fileTABLA DE DERIVADAS Funciones...

TABLA DE DERIVADAS

Funciones elementales Funciones compuestas Función

f(x) Derivada

f '(x) Función

f(u) con u = u(x) Derivada

f '(x) = f '(u).u'(x) f(x) = k f '(x) = 0 f(x) = x f '(x) = 1

pxxf =)( p∈R 1)(' −= ppxxf puuf =)( p∈R ' )(' 1upuuf p−=

xxf ln)( = xxf 1)(' = uxf ln)( = u

uxf ')(' =

xxf alog)( = axxf

ln1)(' = uxf alog)( = au

uxfln

')(' =

xexf =)( xexf =)( ueuf =)( ' )(' ueuf u=xaxf =)( aaxf x ln)( = uauf =)( ' ln)(' uaauf u=

)()()( xhxgxf = )(' )(ln)()(')( )()( )(1)( xhxgxgxgxgxhxf xhxh += −

f(x) = sen x f '(x) = cos x f(x) = sen u f '(x) = cos u u' f(x) = cos x f '(x) = − sen x f(x) = cos u f '(x) = − sen u u'

xxf tg)( = ==x

xf 2cos1)('

x2tg1+=

uxf tg)( = ==u

uxf 2cos')('

' )tg1( 2 uu+=

xxf arcsen)( = 211)('

xxf

−= uxf arcsen)( = 21

')('u

uxf−

=

xxf arccos)( = 211)('x

xf−

−= uxf arccos)( = 21

')('u

uxf−

−=

xxf arctg)( = 211)('x

xf+

= uxf arctg)( = 21')('u

uxf+

=

f(x) = sh x f '(x) = ch x f(x) = sh u f '(x) = ch u u' f(x) = ch x f '(x) = sh x f(x) = ch u f '(x) = sh u u'

xxf th )( = ==x

xf 2ch1)('

x2th1−=

uxf th )( = ==u

uxf 2ch')('

' )th1( 2 uu−=

xxf sh arg)( = 211)('

xxf

+= uxf sh arg)( = 21

')('u

uxf+

=

xxf ch arg)( = 11)('2 −

=x

xf uxf ch arg)( = 1')('

2 −=

uuxf

xxf th arg)( = 21

1)('x

xf−

= uxf th arg)( =

21')('u

uxf−

=