TABELLA DERIVATE FONDAMENTALI e DERIVATE di …dispense\derivate fondamentali e derivate... ·...

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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 ) x n nx n !1 f ( x ) [ ] n n f ( x ) [ ] n !1 " f '( x ) y = sin 2 x ! y ' = 2 sin x cos x es :. x = x 1 2 1 2 x ! 1 2 = 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 ( ) " 2 x cos x ! sin x cos f x () ! sin f x () " f ' x () y = cos 2e x ( ) ! y ' = " sin 2e x ( ) # 2e x tgx 1 cos 2 x = 1 + tg 2 x tg f x () 1 cos 2 x ! f ' x () oppure 1+tg 2 f x () " # $ % ! f ' x () y = tg(2 x + ! 3 ) " y ' = 1 cos 2 (2 x + ! 3 ) # 2 cotgx ! 1 sin 2 x = !(1 + cotg 2 x ) cotg f ( x ) ! 1 sin 2 f ( x ) " f '( x ) arcsenx 1 1 ! x 2 arcsen f ( x ) 1 1 ! f ( x ) [ ] 2 " f ' x () y = arcsen( x 2 ) ! y ' = 1 1 " ( x 2 ) 2 # 2 x arcosx ! 1 1 ! x 2 arcos f ( x ) ! 1 1 ! f ( x ) [ ] 2 " f ' x () artgx 1 1 + x 2 artgf ( x ) 1 1 + f ( x ) [ ] 2 ! f '( x ) y = artg(ln x ) ! y ' = 1 1 + ln 2 x " 1 x arcotgx ! 1 1 + x 2 arcotgf ( x ) ! 1 1 + f ( x ) [ ] 2 " f '( x ) a x a x ln a a f ( x ) a f ( x ) ln a ! f '( x ) e x e x e f ( x ) e f ( x ) f '( x ) y = e sin x ! y ' = e sin x " cos x log a x 1 x ln a = 1 x log a e log a 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 x 2 + 1 2 x

Transcript of TABELLA DERIVATE FONDAMENTALI e DERIVATE di …dispense\derivate fondamentali e derivate... ·...

Page 1: TABELLA DERIVATE FONDAMENTALI e DERIVATE di …dispense\derivate fondamentali e derivate... · scuola internazionale europea statale “Altiero Spinelli” – Torino – a.s. 2007-2008

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