Formule derivate fondamentali e derivate funzioni composte

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scuola internazionale europea statale “Altiero Spinelli” – Torino – a.s. 2007-2008

TABELLA DERIVATE FONDAMENTALI e DERIVATE di FUNZIONI COMPOSTE

y(x)

y'(x)

y(x) = g f (x)

[

]

y'(x) = g' f (x)

[

]

! f '(x)

esempio

k 0

x 1

kx k k ! f (x) k ! f '(x)

xn nxn!1

_[

f (x)

_]

n n f (x)

_[

_]

n!1" f '(x) y = sin2x ! y ' = 2 sin x cos x

es :. x = x12 1 2x ! 12₌ 1 2 x f (x) 1 2 f (x)! f '(x) y = ln x ! y ' = 1 2 ln x " 1 x

sin x cos x sin f x

( )

cos f x

( )

! f ' x

( )

y = sin x

( )

2 ! y ' = cos x

( )

2 " 2x

cos x ! sin x cos f x

( )

! sin f x

( )

" f ' x

( )

y = cos 2e

( )

x ! y ' = " sin 2e

( )

x # 2ex

tgx 1 cos2x= 1+ tg 2_x tg f x

_{( )}

1 cos2x! f ' x

( )

oppure 1+tg 2_{f x}

( )

" # $%! f ' x

( )

y = tg(2x + ! 3) " y ' = 1 cos2(2x +! 3) # 2 cotgx ! 1 sin2x= !(1+ cotg 2_x) _{cotg f (x)} _! 1 sin2 f (x)" f '(x) arcsenx 1 1 ! x2 arcsen f (x) 1 1 ! f (x)

[

]

2 " f ' x

_{( )}

_{y = arcsen(x}2 ) ! y ' = 1 1 " (x2)2 # 2x arcosx ! 1 1 ! x2 arcos f (x) ! 1 1 ! f (x)

_[

_]

2 " f ' x

_{( )}

artgx 1 1+ x2 artgf (x) 1 1+ f (x)

[

]

2! f '(x) y = artg(ln x) ! y ' = 1 1+ ln2x" 1 x arcotgx ! 1 1+ x2 arcotgf (x) ! 1 1+ f (x)

[

]

2" f '(x) ax axln a af (x) af (x)ln a ! f '(x)

ex ex ef (x) ef (x)f '(x) y = esin x! y ' = esin x" cos x

logax 1 x ln a= 1 xlogae loga f (x) 1 f (x) ln a! f '(x) ln x 1 x ln f (x) 1 f (x)! f '(x) y = ln x 2₊₁

(

)

! y ' = 1 x2+12x