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 nx

n!1 f (x)[ ]n n f (x)[ ]

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

2x! 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 x2( )! y ' = cos x2( ) " 2x

cos x !sin x cos f x( ) !sin f x( ) " f ' x( ) y = cos 2ex( )! y ' = "sin 2ex( ) # 2ex

tgx 1

cos2x

=1+ tg2x 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

sin2 x= !(1+ cotg2x) cotg f (x) !

1

sin2f (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 a

xln a a

f (x) af (x)

ln a ! f '(x)

ex e

x ef (x) e

f (x)f '(x) y = e

sin x! y ' = e

sin x" cos x

logax 1

x ln a=1

xlog

ae 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 +1

2x