Regel

Derivatan av en potensfunktion

För att derivera en potensfunktion

f (x) = x^{n},

där

n

är en konstant, multiplicerar man

x^{n}

med

n

och minskar exponenten med

1 .

Derivera/Förenkla

f (x) = x^{4}

f (x) = x^{2}

f (x) = x^{- 5}

Deriveringsregeln gäller för alla reella $n$ . Ibland kan man dock behöva göra vissa omskrivningar för att kunna använda regeln.

Man kan motivera regeln genom att visa att den exempelvis gäller då $n = 2,$ alltså för funktionen $f (x) = x^{2} .$ Man gör detta med hjälp av derivatans definition.

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

SubstituteII

$f (x + h) = (x + h)^{2}$ , $f (x) = x^{2}$

f^{'} (x) = h \to 0 lim \frac{( x + h ) ^{2} - x ^{2}}{h}

ExpandPosPerfectSquare

Utveckla med första kvadreringsregeln

f^{'} (x) = h \to 0 lim \frac{x ^{2} + 2 x h + h ^{2} - x ^{2}}{h}

SimpTerms

Förenkla termer

f^{'} (x) = h \to 0 lim \frac{2 x h + h ^{2}}{h}

SplitIntoFactors

Dela upp i faktorer

f^{'} (x) = h \to 0 lim \frac{h \cdot 2 x + h \cdot h}{h}

FactorOut

Bryt ut $h$

f^{'} (x) = h \to 0 lim \frac{h ( 2 x + h )}{h}

SimpQuot

Förenkla kvot

f^{'} (x) = h \to 0 lim (2 x + h)

To

$h \to 0$

f^{'} (x) = 2 x

Deriveringsregeln gäller alltså när $n = 2,$ och det går att visa det för alla $n$ också.

Även 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

NumberToExponentOne

$a = a^{1}$

f (x) = x^{1}

Derive

Derivera funktion

f^{'} (x) = D (x^{1})

DeriveMonom

$D (x^{n}) = n x^{n - 1}$

f^{'} (x) = 1 \cdot x^{0}

ExponentZero

$a^{0} = 1$

f^{'} (x) = 1

{{ article.displayTitle }}

Versionen från 30 augusti 2018 kl. 14.35

Derivatan av en potensfunktion

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount}}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount}}
	{{ 'ml-lesson-time-estimation' \| message }}

@@ Rad 3: / Rad 3: @@
 <translate><!--T:2-->
 För att derivera en [[Potensfunktion *Wordlist*|potensfunktion]] $f(x)=x^n,$ där $n$ är en konstant, multiplicerar man $x^n$ med $n$ och minskar exponenten med $1.$</translate>
-<!--
-<jsxgpre id="polynomialDerivative" class="jxgbox jsx-canvas">
-var b=mlg.board([-5.5,1.3,5.5,-1.0],{desktopSize:62});
-//Fönstret MÅSTE vara 11 enheter i bredd för att saker ska bli rätt!
-//Alternativt, skala om tScale så att den ser bredden som 11.
-var tScale = 1.75;
-var posX = -0.6;
-var posY = 0;
-var elements = [];
-var whichFunc = 'X4';
-var step = 6;
-var stepUp = function() {
-	step++;
-};
-var animateX4 = function() {
-	if (step === 0) {
- step = 1;
- exp.moveTo([exp.X() - 0.25*tScale, exp.Y()], 400, {callback:
- function() {
- exp.moveTo([exp.X(), exp.Y() - 0.14*tScale], 400);
- }
- });
- f.moveTo([f.X() - 0.15*tScale, f.Y()], 800);
- x.moveTo([x.X() + 0.22*tScale, x.Y()], 800);
- b.show(expStay);
- expStay.moveTo([expStay.X() + 0.22*tScale, expStay.Y()], 800);
- b.fadeIn(prim, 1000);
- b.fadeIn(expSub, 1000, stepUp);
-	}
-	else if (step === 2) {
- step = 3;
- expSub.moveTo([expSub.X() - 0.5*tScale, expSub.Y()], 800);
- b.fadeOut(expSub, 800);
- b.fadeOut(expStay, 800);
- b.fadeIn(expAfter, 1200, stepUp);
-	}
-	else if (step === 4) {
- step = 5;
- for (var i = 0; i < elements.length; i++) {
- b.fadeOut(elements[i], 800);
- }
- setTimeout(function() {
- stepUp();
- animateX4();
- }, 800);
-	}
-	else if (step === 6) {
- step = 7;
- for (var i = 0; i < elements.length; i++) {
- b.board.removeObject(elements[i]);
- }
- par = b.txt(posX,posY,'(x) = ', {opacity:0, fontsize:tScale});
- par.setAttribute({visible:false, opacity:1});
- f = b.txt(par.X() - 0.665*tScale, par.Y(), 'f', {opacity:0, fontsize:tScale});
- f.setAttribute({visible:false, opacity:1});
- x = b.txt(par.X() + 0.775*tScale, par.Y(), 'x', {opacity:0, fontsize:tScale});
- x.setAttribute({visible:false, opacity:1});
- exp = b.txt(par.X() + 1.01*tScale, par.Y() + 0.14*tScale, '4', {opacity:0, fontsize:tScale});
- exp.setAttribute({visible:false, opacity:1});
- expStay = b.txt(par.X() + 1.00*tScale, par.Y() + 0.16*tScale, '4', {opacity:0, fontsize:tScale});
- expStay.setAttribute({visible:false, opacity:1});
- prim = b.txt(par.X() - 0.613*tScale, par.Y() - 0.015*tScale, "'", {visible:true, opacity:0, fontsize:tScale});
- prim.setAttribute({visible:false, opacity:1});
- expSub = b.txt(par.X() + 1.62*tScale, par.Y() + 0.18*tScale, '- 1', {visible:true, opacity:0, fontsize:tScale});
- expSub.setAttribute({visible:false, opacity:1});
- expAfter = b.txt(par.X() + 1.23*tScale, par.Y() + 0.18*tScale, '3', {visible:true, opacity:0, fontsize:tScale});
- expAfter.setAttribute({visible:false, opacity:1});
- elements = [par, prim, x, f, exp, expStay, expSub, expAfter];
- b.fadeIn(par, 800);
- b.fadeIn(f, 800);
- b.fadeIn(x, 800);
- b.fadeIn(exp, 800, function() {step=0;});
-	}
-};
-var animateX2 = function() {
-	if (step === 0) {
- step = 1;
- exp.moveTo([exp.X() - 0.25*tScale, exp.Y()], 400, {callback:
- function() {
- exp.moveTo([exp.X(), exp.Y() - 0.18*tScale], 400);
- }
- });
- f.moveTo([f.X() - 0.15*tScale, f.Y()], 800);
- x.moveTo([x.X() + 0.22*tScale, x.Y()], 800);
- b.show(expStay);
- expStay.moveTo([expStay.X() + 0.22*tScale, expStay.Y()], 800);
- b.fadeIn(prim, 1000);
- b.fadeIn(expSub, 1000, stepUp);
-	}
-	else if (step === 2) {
- step = 3;
- expSub.moveTo([expSub.X() - 0.5*tScale, expSub.Y()], 800);
- b.fadeOut(expSub, 800);
- b.fadeOut(expStay, 800);
- b.fadeIn(expAfter, 1200, stepUp);
-	}
-	else if (step === 4) {
- b.fadeOut(expAfter, 800, stepUp);
-	}
-	else if (step === 5) {
- step = 6;
- for (var i = 0; i < elements.length; i++) {
- b.fadeOut(elements[i], 800);
- }
- setTimeout(function() {
- stepUp();
- animateX2();
- }, 800);
-	}
-	else if (step === 7) {
- step = 8;
- for (var i = 0; i < elements.length; i++) {
- b.board.removeObject(elements[i]);
- }
- par = b.txt(posX,posY,'(x) = ', {opacity:0, fontsize:tScale});
- par.setAttribute({visible:false, opacity:1});
- f = b.txt(par.X() - 0.665*tScale, par.Y(), 'f', {opacity:0, fontsize:tScale});
- f.setAttribute({visible:false, opacity:1});
- x = b.txt(par.X() + 0.775*tScale, par.Y(), 'x', {opacity:0, fontsize:tScale});
- x.setAttribute({visible:false, opacity:1});
- exp = b.txt(par.X() + 1.01*tScale, par.Y() + 0.18*tScale, '2', {opacity:0, fontsize:tScale});
- exp.setAttribute({visible:false, opacity:1});
- expStay = b.txt(par.X() + 1.00*tScale, par.Y() + 0.18*tScale, '2', {opacity:0, fontsize:tScale});
- expStay.setAttribute({visible:false, opacity:1});
- prim = b.txt(par.X() - 0.613*tScale, par.Y() - 0.015*tScale, "'", {visible:true, opacity:0, fontsize:tScale});
- prim.setAttribute({visible:false, opacity:1});
- expSub = b.txt(par.X() + 1.62*tScale, par.Y() + 0.18*tScale, '- 1', {visible:true, opacity:0, fontsize:tScale});
- expSub.setAttribute({visible:false, opacity:1});
- expAfter = b.txt(par.X() + 1.23*tScale, par.Y() + 0.18*tScale, '1', {visible:true, opacity:0, fontsize:tScale});
- expAfter.setAttribute({visible:false, opacity:1});
- elements = [par, prim, x, f, exp, expStay, expSub, expAfter];
- b.fadeIn(par, 800);
- b.fadeIn(f, 800);
- b.fadeIn(x, 800);
- b.fadeIn(exp, 800, function() {step=0;});
-	}
-};
-var animateXM5 = function() {
-	if (step === 0) {
- step = 1;
- exp.moveTo([exp.X() - 0.26*tScale, exp.Y()], 400, {callback:
- function() {
- exp.moveTo([exp.X(), exp.Y() - 0.14*tScale], 400);
- }
- });
- f.moveTo([f.X() - 0.15*tScale, f.Y()], 800);
- x.moveTo([x.X() + 0.36*tScale, x.Y()], 800);
- b.show(expStay);
- expStay.moveTo([expStay.X() + 0.35*tScale, expStay.Y()], 800);
- b.fadeIn(prim, 1000);
- b.fadeIn(expSub, 1000, stepUp);
-	}
-	else if (step === 2) {
- step = 3;
- expSub.moveTo([expSub.X() - 0.5*tScale, expSub.Y()], 800);
- b.fadeOut(expSub, 800);
- b.fadeOut(expStay, 800);
- b.fadeIn(expAfter, 1200, stepUp);
-	}
-	else if (step === 4) {
- step = 5;
- for (var i = 0; i < elements.length; i++) {
- b.fadeOut(elements[i], 800);
- }
- setTimeout(function() {
- stepUp();
- animateXM5();
- }, 800);
-	}
-	else if (step === 6) {
- step = 7;
- for (var i = 0; i < elements.length; i++) {
- b.board.removeObject(elements[i]);
- }
- par = b.txt(posX,posY,'(x) = ', {opacity:0, fontsize:tScale});
- par.setAttribute({visible:false, opacity:1});
- f = b.txt(par.X() - 0.665*tScale, par.Y(), 'f', {opacity:0, fontsize:tScale});
- f.setAttribute({visible:false, opacity:1});
- x = b.txt(par.X() + 0.78*tScale, par.Y(), 'x', {opacity:0, fontsize:tScale});
- x.setAttribute({visible:false, opacity:1});
- exp = b.txt(par.X() + 1.09*tScale, par.Y() + 0.14*tScale, '\\text{-}5', {opacity:0, fontsize:tScale});
- exp.setAttribute({visible:false, opacity:1});
- expStay = b.txt(par.X() + 1.09*tScale, par.Y() + 0.16*tScale, '\\text{-}5', {opacity:0, fontsize:tScale});
- expStay.setAttribute({visible:false, opacity:1});
- prim = b.txt(par.X() - 0.613*tScale, par.Y() - 0.015*tScale, "'", {visible:true, opacity:0, fontsize:tScale});
- prim.setAttribute({visible:false, opacity:1});
- expSub = b.txt(par.X() + 1.91*tScale, par.Y() + 0.16*tScale, '- 1', {visible:true, opacity:0, fontsize:tScale});
- expSub.setAttribute({visible:false, opacity:1});
- expAfter = b.txt(par.X() + 2.14*tScale, par.Y() + 0.16*tScale, '\\text{-}6', {visible:true, opacity:0, fontsize:tScale});
- expAfter.setAttribute({visible:false, opacity:1});
- elements = [par, prim, x, f, exp, expStay, expSub, expAfter];
- b.fadeIn(par, 800);
- b.fadeIn(f, 800);
- b.fadeIn(x, 800);
- b.fadeIn(exp, 800, function() {step=0;});
-	}
-};
-mlg.af("polynomialDifferentiation.X2", function () {
-	whichFunc = 'X2';
-	step = 7;
-	animateX2();
-});
-mlg.af("polynomialDifferentiation.X4", function () {
-	whichFunc = 'X4';
-	step = 6;
-	animateX4();
-});
-mlg.af("polynomialDifferentiation.XM5", function () {
-	whichFunc = 'XM5';
-	step = 6;
-	animateXM5();
-});
-mlg.af("polynomialDifferentiation.animate", function () {
-	if (whichFunc === 'X4') {
- animateX4();
-	}
-	else if (whichFunc === 'X2') {
- animateX2();
-	}
-	else if (whichFunc === 'XM5') {
- animateXM5();
-	}
-});
-//Börja med x^4
-mlg.cf("polynomialDifferentiation.X4");
-mlg.cf("polynomialDifferentiation.animate");
-</jsxgpre>
--->
 <jsxgpre id="polynomialDerivative" class="jxgbox jsx-canvas">
@@ Rad 304: / Rad 28: @@
 	if (step === 0) {
  step = 1;
- exp.moveTo([exp.X() - 0.25*tScale, exp.Y()], 400);
+ exp.moveTo([exp.X() - 0.3*tScale, exp.Y()], 400);
  setTimeout(function() {
- exp.moveTo([exp.X(), exp.Y() - 0.14*tScale], 400);
+ exp.moveTo([exp.X(), par.Y()], 400);
  },400);
- f.moveTo([f.X() - 0.15*tScale, f.Y()], 800);
+ f.moveTo([f.X() - 0.1*tScale, f.Y()], 800);
  x.moveTo([x.X() + 0.22*tScale, x.Y()], 800);
  b.show(expStay);
@@ Rad 342: / Rad 66: @@
  }
- par = b.txt(posX,posY,'(x) = ', {opacity:0, fontsize:tScale});
+ par = b.txt(posX,posY,'(x) = ', {opacity:0,transitionDuration:0, fontsize:tScale});
- f = b.txt(par.X() - 0.665*tScale, par.Y(), 'f', {opacity:0, fontsize:tScale});
+ f = b.txt(par.X() - 0.665*tScale, par.Y(), 'f', {opacity:0,transitionDuration:0, fontsize:tScale});
- x = b.txt(par.X() + 0.775*tScale, par.Y(), 'x', {opacity:0, fontsize:tScale});
+ x = b.txt(par.X() + 0.775*tScale, par.Y(), 'x', {opacity:0,transitionDuration:0, fontsize:tScale});
- exp = b.txt(par.X() + 1.01*tScale, par.Y() + 0.14*tScale, '4', {opacity:0, fontsize:tScale});
+ exp = b.txt(par.X() + 1.01*tScale, par.Y() + 0.18*tScale, '4', {opacity:0,transitionDuration:0, fontsize:tScale});
- expStay = b.txt(par.X() + 1.00*tScale, par.Y() + 0.16*tScale, '4', {opacity:0, fontsize:tScale});
+ expStay = b.txt(par.X() + 1.00*tScale, par.Y() + 0.18*tScale, '4', {opacity:0,transitionDuration:0, fontsize:tScale});
- prim = b.txt(par.X() - 0.613*tScale, par.Y() - 0.015*tScale, "'", {opacity:0, fontsize:tScale});
+ prim = b.txt(par.X() - 0.54*tScale, par.Y() + 0.1*tScale, "'", {opacity:0,transitionDuration:0, fontsize:tScale*0.7});
- expSub = b.txt(par.X() + 1.62*tScale, par.Y() + 0.18*tScale, '- 1', {opacity:0, fontsize:tScale});
+ expSub = b.txt(par.X() + 1.62*tScale, par.Y() + 0.18*tScale, '- 1', {opacity:0,transitionDuration:0, fontsize:tScale});
- expAfter = b.txt(par.X() + 1.23*tScale, par.Y() + 0.18*tScale, '3', {opacity:0, fontsize:tScale});
+ expAfter = b.txt(par.X() + 1.23*tScale, par.Y() + 0.18*tScale, '3', {opacity:0,transitionDuration:0, fontsize:tScale});
  elements = [par, prim, x, f, exp, expStay, expSub, expAfter];
@@ Rad 369: / Rad 93: @@
 	if (step === 0) {
  step = 1;
- exp.moveTo([exp.X() - 0.25*tScale, exp.Y()], 400);
+ exp.moveTo([exp.X() - 0.3*tScale, exp.Y()], 400);
  setTimeout(function() {
- exp.moveTo([exp.X(), exp.Y() - 0.18*tScale], 400);
+ exp.moveTo([exp.X(), par.Y()], 400);
  },400);
- f.moveTo([f.X() - 0.15*tScale, f.Y()], 800);
+ f.moveTo([f.X() - 0.1*tScale, f.Y()], 800);
  x.moveTo([x.X() + 0.22*tScale, x.Y()], 800);
  b.show(expStay);
@@ Rad 395: / Rad 119: @@
 	else if (step === 5) {
  step = 6;
- //for (var i = 0; i < elements.length; i++) {
+ b.hide(elements, 800);
- //b.hide(elements[i], 800);
- b.hide(elements, 800);
- //}
  setTimeout(function() {
  stepUp();
@@ Rad 410: / Rad 131: @@
  }
- par = b.txt(posX,posY,'(x) = ', {opacity:1, fontsize:tScale});
+ par = b.txt(posX,posY,'(x) = ', {opacity:0,transitionDuration:0, fontsize:tScale});
- f = b.txt(par.X() - 0.665*tScale, par.Y(), 'f', {opacity:0, fontsize:tScale});
+ f = b.txt(par.X() - 0.665*tScale, par.Y(), 'f', {opacity:0,transitionDuration:0, fontsize:tScale});
- x = b.txt(par.X() + 0.775*tScale, par.Y(), 'x', {opacity:0, fontsize:tScale});
+ x = b.txt(par.X() + 0.775*tScale, par.Y(), 'x', {opacity:0,transitionDuration:0, fontsize:tScale});
- exp = b.txt(par.X() + 1.01*tScale, par.Y() + 0.18*tScale, '2', {opacity:0, fontsize:tScale});
+ exp = b.txt(par.X() + 1.01*tScale, par.Y() + 0.18*tScale, '2', {opacity:0,transitionDuration:0, fontsize:tScale});
- expStay = b.txt(par.X() + 1.00*tScale, par.Y() + 0.18*tScale, '2', {opacity:0, fontsize:tScale});
+ expStay = b.txt(par.X() + 1.00*tScale, par.Y() + 0.18*tScale, '2', {opacity:0,transitionDuration:0, fontsize:tScale});
- prim = b.txt(par.X() - 0.613*tScale, par.Y() - 0.015*tScale, "'", {opacity:0, fontsize:tScale});
+ prim = b.txt(par.X() - 0.54*tScale, par.Y() + 0.1*tScale, "'", {opacity:0,transitionDuration:0, fontsize:tScale*0.7});
- expSub = b.txt(par.X() + 1.62*tScale, par.Y() + 0.18*tScale, '- 1', {opacity:0, fontsize:tScale});
+ expSub = b.txt(par.X() + 1.62*tScale, par.Y() + 0.18*tScale, '- 1', {opacity:0,transitionDuration:0, fontsize:tScale});
- expAfter = b.txt(par.X() + 1.23*tScale, par.Y() + 0.18*tScale, '1', {opacity:0, fontsize:tScale});
+ expAfter = b.txt(par.X() + 1.23*tScale, par.Y() + 0.18*tScale, '1', {opacity:0,transitionDuration:0, fontsize:tScale});
  elements = [par, prim, x, f, exp, expStay, expSub, expAfter];
@@ Rad 437: / Rad 158: @@
 	if (step === 0) {
  step = 1;
- exp.moveTo([exp.X() - 0.26*tScale, exp.Y()], 400);
+ exp.moveTo([exp.X() - 0.3*tScale, exp.Y()], 400);
  setTimeout(function() {
- exp.moveTo([exp.X(), exp.Y() - 0.14*tScale], 400);
+ exp.moveTo([exp.X(), par.Y()*tScale], 400);
  },400);
- f.moveTo([f.X() - 0.15*tScale, f.Y()], 800);
+ f.moveTo([f.X() - 0.1*tScale, f.Y()], 800);
  x.moveTo([x.X() + 0.36*tScale, x.Y()], 800);
  b.show(expStay);
@@ Rad 473: / Rad 194: @@
  }
- par = b.txt(posX,posY,'(x) = ', {opacity:0, fontsize:tScale});
+ par = b.txt(posX,posY,'(x) = ', {opacity:0,transitionDuration:0, fontsize:tScale});
- f = b.txt(par.X() - 0.665*tScale, par.Y(), 'f', {opacity:0, fontsize:tScale});
+ f = b.txt(par.X() - 0.665*tScale, par.Y(), 'f', {opacity:0,transitionDuration:0, fontsize:tScale});
- x = b.txt(par.X() + 0.78*tScale, par.Y(), 'x', {opacity:0, fontsize:tScale});
+ x = b.txt(par.X() + 0.78*tScale, par.Y(), 'x', {opacity:0,transitionDuration:0, fontsize:tScale});
- exp = b.txt(par.X() + 1.09*tScale, par.Y() + 0.14*tScale, '\\text{-}5', {opacity:0, fontsize:tScale});
+ exp = b.txt(par.X() + 1.09*tScale, par.Y() + 0.18*tScale, '\\text{-}5', {opacity:0,transitionDuration:0, fontsize:tScale});
- expStay = b.txt(par.X() + 1.09*tScale, par.Y() + 0.16*tScale, '\\text{-}5', {opacity:0, fontsize:tScale});
+ expStay = b.txt(par.X() + 1.09*tScale, par.Y() + 0.18*tScale, '\\text{-}5', {opacity:0,transitionDuration:0, fontsize:tScale});
- prim = b.txt(par.X() - 0.613*tScale, par.Y() - 0.015*tScale, "'", {opacity:0, fontsize:tScale});
+ prim = b.txt(par.X() - 0.54*tScale, par.Y() + 0.1*tScale, "'", {opacity:0,transitionDuration:0, fontsize:tScale*0.7});
- expSub = b.txt(par.X() + 1.91*tScale, par.Y() + 0.16*tScale, '- 1', {opacity:0, fontsize:tScale});
+ expSub = b.txt(par.X() + 1.91*tScale, par.Y() + 0.18*tScale, '- 1', {opacity:0,transitionDuration:0, fontsize:tScale});
- expAfter = b.txt(par.X() + 1.44*tScale, par.Y() + 0.16*tScale, '\\text{-}6', {opacity:0, fontsize:tScale});
+ expAfter = b.txt(par.X() + 1.44*tScale, par.Y() + 0.18*tScale, '\\text{-}6', {opacity:0,transitionDuration:0, fontsize:tScale});
@@ Rad 531: / Rad 252: @@
 mlg.cf("polynomialDifferentiation.X4");
 </jsxgpre>
 <div class='jsx-btn-container'>
@@ Rad 542: / Rad 262: @@
 <jsxbtn style="width:25%;" onclick='mlg.cf("polynomialDifferentiation.XM5")'>$f(x) = x^{\text{-}5}$</jsxbtn>
 </div>
 <translate><!--T:8-->