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Gør differentiering til en leg

I denne artikel ser vi nærmere på differentialregning – altså hvordan man differentiere. Artiklen henvender sig primært til de studerende som har matematik på niveau A eller niveau B på de gymnasielle uddannelser. Det er en kunst at mestrer differentialregningen, og emnet er også en de faldgrupper som de studerende som oftest har svært ved når de går til eksamen.

For at blive god til differentialregning er det vigtigt, at man kender sine regneregler. Det er alfa og omega, hvis man gerne vil opnå resultater. I denne artikel ser vi først på en introduktion til differentialregningen og herefter de afledede funktioner i forbindelse med differentialregningen. Efterfølgende vil vi tage udgangspunkt i en eksamensopgave, så du som studerende lærer hvordan fremgangsmåden ved løsningen af opgaven skal gribes an.

Introduktion til differentialregning

Når man skal differentiere en funktion er det første man skal være opmærksom på, at en funktion der differentieres angives ved f´(x) = (udtales f mærke af x).

Den nemmeste måde at gribe det an på er, at man differentiere led for led. Lad os tage et eksempel. Herunder er forskriften for en funktion:

formel for en funktion

 

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Vi skal altså bestemme f´(x). Der gælder generelt for en funktion der adderes at:

forsteled og andet led

Alt efter hvor mange led (adskilt af plus eller minus), der er i funktionsudtrykket.

Det første led i funktionsforskriften er 3x3. Når vi skal differentiere et led, så er det vigtigt at vide, at der generelt gælder, at x opløftet i et tal n (dvs. xn) differentieret giver

her vises hvordan man differentiere - formlen er hermed givet

Vi har nu differentieret det første led i funktionsforskriften. Vi differentiere nu andet led i funktionsforskriften 5x2 på samme måde som før.

her vises hvordan man differentierer

Det næste led der skal differentieres er det tredje led i funktionsforskriften som er 10x. Dette er en konstant (10) ganget på x, og differentieret giver det derfor konstanten, altså 10.

Det sidste led er det fjerde i funktionsforskriften som er 4. Når man differentierer en konstant giver det altid 0.

Vi har nu differentieret alle fire led i funktionsforskriften. Vi kan derfor liste den differentieret funktionsforskrift. Det er gjort herunder;

den differentierede funtion listes nu

De afledede funktioner i differentialregning

her vises de afledte funktioner i differential regning

Herunder vil vi berør de afledede funktioner i differentialregningen. Hvis du har/får styr på disse, så er du kommet langt, og du vil med sikkerhed gøre differentialregningen til en leg ved at øve dig. Når du først har fået forståelsen, vil du opleve at det faktisk ikke er så svært, og pludselig er ordet differentialregning ikke længere et fy ord.

I introduktionen har vi allerede berørt flere af de afledede funktioner. Vi har nemlig allerede set på regelnr. 2, 3 og 4 jf. ovenstående tabel. Vi vil herunder komme med eksempler for hver af de basale regneregler.

Afledte funktion nummer 1 og 2

Vi har herunder listet en funktion, som vi ønsker at differentiere:

f(x) = x + 345

Hvis du er hjemvendt i matematikkens univers, så vil du på nuværende tidspunkt allerede vide, at funktionen i vores eksempel er en lineær funktion. Hvis man skal differentiere den, så ville du skulle benytte dig af regel nummer 1 og 2, da vi skal differentiere x og differentiere 345.

Vi differentiere hvert led for sig. Vi differentiere første led og herefter andet led. Det første der skal differentieres er x. Vi kan ud fra tabellens afledte funktion nummer 1 se, at x differentieret giver 1. Det vises herunder:

her vises hvordan man differentiere den afledete funktion nummer 1 og 2

Vi har nu tilbage at differentiere det sidste led, som er en konstant udtrykt ved 345. Når en konstant differentieres bliver det altid = 0. Det vises herunder:

afledte funktion 1 og 2

Nu har vi differentieret hvert led for sig, og vi har nu bare tilbage at liste den differentieret funktion, hvilket er vist herunder:

resultatet på den afledte funktionen 1 og 2

Afledte funktion nummer 3 og 4

Disse er gennemgået og forklaret i selve introduktionen længere oppe.

 

 

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Afledte funktion nummer 5 og 6

Herunder har vi listet en funktion, som der skal differentieres:

her vises hvordan man differentiere den afledte funktion 5 og 6

Som tidligere nævnt så differentiere vi hvert led for sig. Så først differentiere vi første led ved hjælp af afledte funktion nummer 5:

her er resultatet af den afledte funktion 5 og 6

det endelige resultat af den afledte

Læg mærke til med denne afledte funktion, at det ikke gør nogen forskel hvis vi ganger en konstant på i nævneren. Vi har nu differentieret det første led. Vi fortsætter med at differentiere det andet led i vores funktion. Her skal vi anvende den afledte funktion nummer 6 ved differentieringen:

her er det andet led i differentieringen vi kigger på

Vi har nu differentieret det andet led i vores funktion. Det man skal ligge mærke til her, det er at vi har at gøre med en potens funktion der skal differentieres. Det gælder at når man differentierer en potensfunktion, så vil potensfunktionen blive ”stående” og man ganger med ln til den foranstående konstant i potensfunktionen. Vi har nu tilbage at liste den differentierede funktion. Det vises herunder:

her er resultatet af alle afledte led

Afledte funktion nummer 7 og 8

Herunder har vi igen vist en funktion som vi ønsker at differentierer:

her ser vi på den afledte funktion af e opløftet i x

Igen differentierer vi hvert led for sig. Første led differentieres ved hjælp af afledte funktion nummer 7. Differentieringen foretages herunder:

her vises selve den afledte af ex

Vi har nu differentieret ex, og når man differentierer ex giver det altid ex. Det næste led differentieres ved hjælp af afledte funktion nummer 8. Differentieringen foretages herunder:

her vises resultatet

Vi har nu fået differentieret begge led, og det sidste der mangler er, at få liste vores differentieret funktion. Det har vi vist herunder:

Afledte funktion nummer 9 og 10

Som i de andre eksempler har vi herunder opstillet en funktion som vi ønsker at differentierer:

19

Vi differentierer hvert led for sig, ligesom vi har gjort i de tidligere eksempler. Vi anvender afledte funktion nummer 9 til at differentierer det første led:

20

Vi har nu differentieret det første led. Hvis der ganges en konstant på ”roden af x”, så kan man sige, at man deler konstanten med 2, og kommer det op i tælleren, og det omvendte sker hvis vi dividere en konstant med ”roden af x”, så ganges der med 2 og tallet stilles foran ”én divideret med roden af x” i nævneren. Nu har vi differentieret første led, og mangler at differentierer det sidste led. Det gøres ved hjælp af afledte funktion nummer 10 herunder:

21

Vi har differentieret andet led i vores funktion. Det man skal være opmærksom på, når man differentiere ln(x) er, at ln(x) isoleret set bliver én delt med x. Hvis man ganger en konstant på ln(x) bliver det til konstanten delt med x (konstanten står i tælleren). Hvis vi ønsker at differentiere ln(x) delt med en konstant, så bliver det til ”én over konstanten ganget med x”.

Det sidste vi mangler er, at opstille den differentieret funktion. Det har vi gjort herunder:

22

Regneregler for differentialkvotienter

I vores gennemgang af de afledte funktioner i differentialregningen har vi kun set på funktioner hvor der adderes eller hvor der ganges en konstant på i den funktion vi ønsker at differentierer. Man skal dog være opmærksom på, at der findes forskellige regneregler alt efter om man har sammensat funktion hvor der adderes, subtraheres, multipliceres eller divideres. Vi herunder gennemgå de forskellige regneregler.

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