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

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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 = sin2 x! 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 1cos

    2x

    =1+ tg2x tg f x( ) 1

    cos2x! f ' x( ) oppure 1+tg2 f x( )"#

    $%! f ' x( )

    y = tg(2x +!

    3)" y ' =

    1

    cos2(2x +

    !

    3)

    # 2

    cotgx ! 1sin2 x

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

    2f (x)

    " f '(x)

    arcsenx 1

    1! x2

    arcsen f (x) 1

    1! f (x)[ ]2" f ' x( ) y = arcsen(x2 )! y ' = 1

    1" (x2)2# 2x

    arcosx !1

    1! x2

    arcos f (x) !1

    1! f (x)[ ]2" f ' x( )

    artgx 11+ x

    2 artgf (x)

    1

    1+ f (x)[ ]2! f '(x) y = artg(ln x)! y ' = 1

    1+ ln2x"1

    x

    arcotgx ! 11+ x

    2 arcotgf (x) !

    1

    1+ f (x)[ ]2" f '(x)

    ax ax ln a a f (x) a f (x) ln a ! f '(x)

    ex ex e f (x) e f (x) f '(x) y = esin x ! y ' = esin 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 x2 +1( )! y ' = 1

    x2 +1

    2x