1. Deriveringsregler för potensfunktioner
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Kapitel 3
1.
Deriveringsregler för potensfunktioner
Innehållet handlar om deriveringsregler för potensfunktioner. Den förklarar hur man kan använda dessa regler för att derivera olika typer av funktioner, inklusive de som innehåller bråk och rotuttryck. Sidan ger en detaljerad genomgång av hur man kan omvandla dessa uttryck till potensform och sedan använda deriveringsreglerna för att hitta derivatan. Dessutom innehåller sidan flera exempel och steg-för-steg-lösningar för att hjälpa läsaren att förstå och tillämpa koncepten. Detta är en utmärkt lektionen för studenter som vill förbättra sina färdigheter i derivering och förstå hur man kan tillämpa dessa tekniker på olika typer av funktioner.
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Deriveringsregler för potensfunktioner
Sida av 7
I den här lektionen går vi igenom följande begrepp:
- Derivatan av en potensfunktion
- Beräkna derivatans värde med deriveringsregler
- Skriv om och derivera potensfunktion
Förkunskaper
Koncept
Derivera funktioner
Begreppet derivata kan syfta på lutningen i en viss punkt, men det kan också vara en funktion som beskriver hur derivatan beror på olika x-värden. När man deriverar en funktion f(x) får man derivatan
f′(x)
som också är en funktion. Genom att sätta in ett x-värde i f′(x) kan man beräkna derivatans värde för detta x. Exempelvis innebär f′(2) att man beräknar derivatans värde för f(x) när x=2. Förutom skrivsättet f′(x) kan derivatan till en funktion f även skrivas som D(f(x)) som utläses "derivatan av f(x)": f′(x)⇔D(f(x)).
Detta är användbart om man vill se vilken funktion som deriveras. T.ex. kan derivatan av f(x)=4x2 skrivas D(4x2). För att derivera en funktion kan man använda derivatans definition eller deriveringsregler. Deriveringsreglerna talar om hur olika typer av funktioner deriveras utan att man behöver använda definitionen för derivata.Regel
Derivatan av en potensfunktion
För att derivera en potensfunktion f(x)=xn, där n är en konstant, multiplicerar man xn med n och minskar exponenten med 1.
Deriveringsregeln gäller för alla reella n.
Regel
D(xn)=nxn−1Man kan motivera regeln genom att visa att den exempelvis gäller då n=2, alltså för funktionen f(x)=x2. Man gör detta med hjälp av derivatans definition.
f′(x)=h→0limhf(x+h)−f(x)
SubstituteII
f(x+h)=(x+h)2 och f(x)=x2
f′(x)=h→0limh(x+h)2−x2
ExpandPosPerfectSquare
Utveckla med första kvadreringsregeln
f′(x)=h→0limhx2+2xh+h2−x2
SimpTerms
Förenkla termer
f′(x)=h→0limh2xh+h2
SplitIntoFactors
Dela upp i faktorer
f′(x)=h→0limhh⋅2x+h⋅h
FactorOut
Bryt ut h
f′(x)=h→0limhh(2x+h)
SimpQuot
Förenkla kvot
f′(x)=h→0lim(2x+h)
To
h→0
f′(x)=2x
Regel
D(x)=1Även funktionen f(x)=x går att derivera med deriveringsregeln för potensfunktioner, eftersom x är en potens med graden 1. Ofta brukar man dock använda en snabbare väg, nämligen regeln som säger att D(x)=1, som härleds här.
f(x)=x
f(x)=x1
Derive
Derivera funktion
f′(x)=D(x1)
DeriveMonom
D(xn)=nxn−1
f′(x)=1⋅x0
ExponentZero
a0=1
f′(x)=1
Exempel
Derivera potensfunktionerna
Derivera potensfunktionerna.
a
f(x)=x7
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b
g(x)=x2,3
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c
h(x)=x−10
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d
k(x)=x
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Ledtråd
a När du deriverar en potensfunktion multiplicerar du funktionsuttrycket med exponenten och minskar sedan exponenten med 1.
b Använd deriveringsregeln för potensfunktioner.
c Använd deriveringsregeln för potensfunktioner.
d Använd deriveringsregeln för potensfunktioner.
Lösning
a Detta är fyra funktioner som kan skrivas på formen xn, så vi använder deriveringsregeln för potensfunktioner. Den säger att en funktion på formen f(x)=xn har derivatan f′(x)=nxn−1. Man läser alltså av exponenten och sätter den som koefficient framför funktionen och minskar sedan exponenten med 1. Vi börjar med f(x)=x7.
f(x)=x7
Derive
Derivera funktion
f′(x)=D(x7)
DeriveMonom
D(xn)=nxn−1
f′(x)=7x7−1
SubTerm
Subtrahera term
7x6
b För g(x)=x2.3 är exponenten inte ett heltal, men det påverkar inte hur man använder regeln. Man får derivatan
g′(x)=2,3x2,3−1=2,3x1,3.
c Nästa funktion har en negativ exponent, men vi kan fortfarande använda samma regel. Derivatan av h(x)=x−10 är alltså
h′(x)=−10x−10−1=−10x−11.
d Till sist har vi k(x)=x, som vi skulle kunna derivera på samma sätt genom att skriva om x som x1. Men vi kan lika gärna använda specialregeln D(x)=1:
k′(x)=1.
Metod
Beräkna derivatans värde med deriveringsregler
Ofta vill man bestämma värdet för en funktions derivata i en specifik punkt. Exempelvis kan man bestämma f′(5) för f(x)=x4. Det innebär att man ska bestämma derivatan när x=5.
1
Derivera funktionen
expand_more Man börjar med att derivera funktionen med lämplig deriveringsregel. I det här fallet har man en potensfunktion så man använder därför deriveringsregeln för potensfunktioner.
2
Sätt in x-värde
expand_more Sedan beräknas derivatans värde genom att man sätter in x-värdet i f′(x).
Derivatans värde är alltså 500 i när x=5.
f′(x)=4x3
Substitute
x=5
f′(5)=4⋅53
CalcPow
Beräkna potens
f′(5)=4⋅125
Multiply
Multiplicera faktorer
f′(5)=500
Metod
Skriv om och derivera potensfunktion
Rotuttryck och bråk där variabeln står i nämnaren kan ibland skrivas om på potensform. T.ex. kan x skrivas om som en potens med exponenten 0,5,
x=x0,5,
och x21 kan skrivas som en potens med negativ exponent:
x21=x−2.
Detta kan utnyttjas för att skriva om vissa funktioner till formen f(x)=xn så att man kan använda deriveringsregeln för potensfunktioner. Detta gäller exempelvis funktionen f(x)=x1.
1
Finns det något rotuttryck som kan skrivas som en potens?
expand_more Om funktionen innehåller ett rotuttryck skriver man om det som en potens med ett bråk eller decimaltal i exponenten. Nämnaren i det här funktionsuttrycket, x, kan skrivas som x0,5, och funktionen kan därför skrivas
f(x)=x0,51.
2
Finns det något bråk som kan skrivas som en potens med negativ exponent?
expand_more Om funktionen är ett bråk på formen xn1 skriver man om det som potensen x−n. I det här fallet får man
f(x)=x0,51=x−0,5.
3
Använd deriveringsregeln för potensfunktioner
expand_more Nu kan man derivera funktionen med deriveringsregeln för potensfunktioner.
När man har kommit så här långt är man egentligen färdig med deriveringen, så det sista steget i metoden behöver inte alltid användas.
4
Skriv eventuellt derivatan på samma form som ursprungsfunktionen
expand_more I vissa fall kan det vara lämpligt att skriva om derivatan så att den står på liknande form som f(x). Det kan till exempel vara om det krävs i uppgiften eller om det blir enklare att räkna vidare med. För det här exemplet skulle man i så fall skriva om derivatan så den står på bråkform och innehåller ett rotuttryck.
I det här fallet kan man alltså skriva derivatan till funktionen f(x)=x1 som
f′(x)=−0,5x−1,5
▼
Skriv om
NegExponentToFrac
a-b=ab1
f′(x)=−0,5⋅x1,51
WriteAsFrac
Skriv i bråkform
f′(x)=−21⋅x3/21
anm=nam
f′(x)=−21⋅x31
SplitIntoFactors
Dela upp i faktorer
f′(x)=−21⋅x2⋅x1
SqrtProd
a⋅b=a⋅b
f′(x)=−21⋅x2⋅x1
SqrtPowToNumber
a2=a
f′(x)=−21⋅x⋅x1
MultFrac
Multiplicera bråk
f′(x)=−2xx1
f′(x)=−2xx1.
Exempel
Derivera en potensfunktion och beräkna derivatans värde
a Differentiate the function.
f(x)=5x1
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b Find 5⋅f′(32). Svara exakt.
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Ledtråd
a Start by rewriting the root expression as a power with a fractional exponent.
b Subsitute 32 for x in the derivative of the function and multiply the result by 5.
Lösning
a The function contains a root expression, rewrite it as a power with a fraction in the exponent. It then can be expressed with a negative exponent.
f(x)=x−1/5
Derive
Derivera funktion
f′(x)=D(x−1/5)
DeriveMonom
D(xn)=nxn−1
f′(x)=−51⋅x−51−1
SubTerm
Subtrahera term
f′(x)=−51⋅x−6/5
MoveRightFacToNumOne
b1⋅a=ba
f′(x)=−5x−6/5
f′(x)=−5x−6/5
▼
Skriv om
MoveNumRight
ba=b1⋅a
f′(x)=−51⋅x−6/5
NegExponentToFrac
a-b=ab1
f′(x)=−51⋅x6/51
anm=nam
f′(x)=−51⋅5x61
SplitIntoFactors
Dela upp i faktorer
f′(x)=−51⋅5x5⋅x1
RootProd
5a⋅b=5a⋅5b
f′(x)=−51⋅5x5⋅5x1
RootPowToNumber
5a5=a
f′(x)=−51⋅x⋅5x1
Multiply
Multiplicera faktorer
f′(x)=−5x5x1
b First multiply the derivative by 5. Then subsitute x=32.
f′(x)=−5x5x1
5f′(x)=−x5x1
Substitute
x=32
5f′(32)=−325321
▼
Beräkna högerled
WritePow
Skriv som potens
5f′(32)=−325251
RootPowToNumber
5a5=a
5f′(32)=−32⋅21
Multiply
Multiplicera faktorer
5f′(32)=−641
Derivera funktionen.
f(x)=x2
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g(x)=x5
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h(x)=x6
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">h<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["6x^5"]}}
p(x)=x9
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">p<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["9x^8"]}}
q(x)=x12
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">q<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["12x^{11}"]}}
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För att derivera en potensfunktion multiplicerar vi funktionsuttrycket, dvs. x^2, med exponenten samt minskar exponenten med 1.
Vi gör på samma sätt igen, dvs. multiplicerar funktionsuttrycket, x^5, med exponenten samt minskar exponenten med 1.
Vi fortsätter på samma sätt.
Vi deriverar på samma sätt ännu en gång.
Vi gör samma sak igen.
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Derivera funktionen.
f(x)=x4
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["4x^3"]}}
g(x)=x105
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">g<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["105x^{104}"]}}
h(x)=x
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">h<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["1"]}}
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Vi använder deriveringsregeln för potensfunktioner. Det betyder att 4:an placeras framför x och att exponenten minskas med 1.
Vi gör på samma sätt. 105 flyttas ner och exponenten minskas med 1.
Derivatan av x är lika med 1 enligt deriveringsregeln för potensfunktioner.
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Derivera funktionen.
y=x−2
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.946332em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["-2x^{-3}"]}}
y=x3,5
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.946332em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["3.5x^{2.5}"]}}
y=x−9,6
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.946332em;vertical-align:-0.19444em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["-9.6x^{-10.6}"]}}
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Vi använder deriveringsregeln för potensfunktioner.
Vi fortsätter på samma sätt och deriverar med deriveringsregeln för potensfunktioner.
Inget nytt. Vi fortsätter som vanligt.
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För funktionen f(x)=x4, bestäm derivatan i följande x-värde.
x=2
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">2<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["32"]}}
x=−100
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord text\"><span class=\"mord\">-<\/span><\/span><span class=\"mord\">1<\/span><span class=\"mord\">0<\/span><span class=\"mord\">0<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["-4000000"]}}
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För att bestämma derivatan måste vi först derivera funktionen.
Genom att sätta in x=2 i derivatan bestämmer vi derivatans värde i punkten med denna x-koordinat.
Derivatan i punkten där x=2 är alltså 32.
Samma sak igen. Vi har redan deriverat funktionen så vi sätter helt enkelt in x=- 100 i derivatan.
Derivatan är - 4 000 000 i punkten där x=- 100.
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Bestäm g′(3) för funktionen.
g(x)=x3
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">g<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["27"]}}
g(x)=x7
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">g<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["5103"]}}
g(x)=x11
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">g<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["649539"]}}
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Vi börjar med att derivera g(x) med deriveringsregeln för potensfunktioner för att bestämma g'(x).
Nu kan vi sätta in x = 3 för att bestämma derivatan för det x-värdet.
Vi gör på samma sätt för den här funktionen och börjar med att bestämma g'(x) med hjälp av deriveringsregeln för potensfunktioner.
Vi sätter sedan in x=3 och beräknar g'(3).
Vi gör på samma sätt ännu en gång. Först deriverar vi funktionen g(x) = x^(11).
Till sist sätter vi in x=3.
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Bestäm f′(3) för funktionen.
f(x)=x4
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["108"]}}
f(y)=y2
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["6"]}}
f(z)=z−1
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord\">3<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["-\\dfrac{1}{9}"]}}
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Vi deriverar funktionen med deriveringsregeln för potensfunktioner och sätter sedan in x=3.
Nu sätter vi in x=3 och beräknar.
f'(3) är alltså lika med 108.
Vi gör på samma sätt och börjar med att derivera och sätter sedan in y=3.
Inget nytt. Vi fortsätter på samma sätt.
Innan vi sätter in z=3 skriver vi om uttrycket till ett bråk för att det ska bli enklare att beräkna.
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Derivera funktionen.
f(x)=x1/2
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10764em;\">f<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\\dfrac{x^{-\\dfrac{1}{2}}}{2}","\\dfrac{1}{2\\sqrt{x}}","\\dfrac{x^{-0.5}}{2}","\\dfrac{1}{2x^{0.5}}"]}}
g(x)=x1/4
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">g<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\\dfrac{x^{-\\dfrac{3}{4}}}{4}","\\dfrac{1}{2x^{\\dfrac{3}{4}}}","\\dfrac{x^{-0.75}}{4}","\\dfrac{1}{4x^{0.75}}"]}}
h(x)=x−1/3
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.001892em;vertical-align:-0.25em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">h<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.751892em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["-\\dfrac{x^{-\\dfrac{4}{3}}}{3}","-\\dfrac{1}{2x^{\\dfrac{4}{3}}}"]}}
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Vi använder deriveringsregeln för potensfunktioner, dvs. vi multiplicerar funktionsuttrycket med exponenten samt minskar exponenten med 1.
Vi använder samma deriveringsregel även här.
Vi gör på samma sätt igen.
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Deriveringsregler för potensfunktioner
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