Differentialregning

Differentiation af simple funktioner og regneregler

Definitioner

f 'x0 :=

h 0lim as

f er kontinuert i x0 x x0

lim f x = f x0

=h

xfhxfh

)()(lim 00

0

Definition

Definitionf er differentiabel i x0

Δy

h

x0

f(x0)

f(x0+ h)

f(x)=x2

(x0+ h , f(x0+ h))

f 'x0 :=

h 0lim as= h

xfhxfh

)()(lim 00

0

Definitionf er differentiabel i x0

Tangentligning:y = f ’(x0)(x – x0) + f(x0)

Regneregler

)(' xfk

)(')(' xgxf

)(')(' xgxf

)(')()(')( xgxfxfxg

)(

)(')()()('2 xg

xgxfxgxf

(kf)'(x)

(f + g)'(x)

(f - g)'(x)

(f g)'(x)

)(' xgf

Bevis for (kf)'(x) =

h

xhxh

))(())((lim

0

h

xfkh

)(lim:)()(

0

h

xfhxfh

))()((lim

0)(' x

h

xhxh

)()(lim

0

h

xfhxfh

)()(lim

0

)(' xfk

fk fk fk fk

k k fk

fk

Bevis for

h

xhxh

))(())((lim

0

)()'( xgf

:)()( x

h

xxhxhxh

))()(()()(lim

0

hh 0

lim

h

xghxg

h

xfhxfhh

)()(lim

)()(lim

00)()( xgxf

)()( xgxf

gf gf gf

gfgf )()( xfhxf ))()(( xghxg

Bevis for (fg)'(x)

:)()( xgf

h

xgxfxfhxgxfhxghxghxfh

)()()()()()()()(lim

0

h

xghxgxfhxgxfhxfh

))()(()()())()((lim

0

)()()(

)()()(

lim0

xfh

xghxghxg

h

xfhxfh

)(lim

)()(lim)(lim

)()(lim

0000xf

h

xghxghxg

h

xfhxfhhhh

h

xgfhxgfh

))(())((lim

0

h

xgxfhxghxfh

)()()()(lim

0

)()(')(lim)('0

xfxghxgxfh

)()()()( xgxfxgxf

)()()()( xgxfxgxf

Bevis for )(' xgf

Fortsættes

hh 0lim h

xhxh

)()(lim

0

hh 0

lim

hh

brøkstregfælles

0lim

0

lim

h

brøkregler

0

lim

h

Trylleri

:)(' x

g

f

)(

)(')()()('2 xg

xgxfxgxf 0g

g

f )(gf

)(gf

)()(

)()(

xgxf

hxghxf

))()(

)()()()((

xghxghxgxfxghxf

h

hxgxfxghxf

xghxg

)()()()(

)()(

1

h

hxgxfxgxfxgxfxghxf

xghxg

)()()()()()()()(

)()(

1

0

)(

limh

foruden

0limh

0limh

h

gxfxg

h

f

xghxg hhhhhh

ireglergrænseværd

000000

lim)(lim)(limlim)(lim)(lim

1

)(')()()(')()(

1xgxfxgxf

xgxg

)(

)(')()()('2 xg

xgxfxgxf

h

xghxgxfxgxfhxf

xghxg

))())(()()())()((

)()(

1

h

gxfxgf

xghxg

)()(

)()(

1

h

gxfxgf

xghxg

)()(

)()(

1

)(' xgf

)(

)(')()()('2 xg

xgxfxgxf 0g