L3 The Power Rule and The Sum and Differences Rules of Derivatives - Part 5
Completion requirements
Unit 2A
Derivatives Part 1
Lesson 3: The Power Rule and The Sum and Differences Rules of Derivatives
The Sum and Difference Rules
| The Sum Rule If two functions are differentiable, then the sum of the functions is also differentiable. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«mo mathvariant=¨italic¨»`«/mo»«mo»=«/mo»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mi»g«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/math» Alternative notation: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» The Difference Rule If two functions are differentiable, then the difference of the functions is also differentiable. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»-«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«mo mathvariant=¨italic¨»`«/mo»«mo»=«/mo»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»-«/mo»«mi»g«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/math» Alternative notation: «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mfenced open=¨[¨ close=¨]¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»-«/mo»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfenced»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»-«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math» |
Find the derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»4«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
In the previous Lesson, the derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»4«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math»
was found using first principles.
Using the sum rule, and applying the power rule, the derivative can also be determined as follows.
The derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»4«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math» is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mn»4«/mn»«/math».
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mi»h«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8722;«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mi»h«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mfenced»«mrow»«mn»4«/mn»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mi»h«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»§#8722;«/mo»«mfenced»«mrow»«mn»4«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mrow»«mi»h«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfrac»«mrow»«mn»4«/mn»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«mi»h«/mi»«mo»+«/mo»«mn»1«/mn»«mo»§#8722;«/mo»«mn»4«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»1«/mn»«/mrow»«mi»h«/mi»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfenced»«mfrac»«mrow»«mn»4«/mn»«mi»h«/mi»«/mrow»«mi»h«/mi»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«munder»«mi»lim«/mi»«mrow»«mi»h«/mi»«mo»§#8594;«/mo»«mn»0«/mn»«/mrow»«/munder»«mfenced»«mn»4«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«/mtd»«/mtr»«/mtable»«/math»
Using the sum rule, and applying the power rule, the derivative can also be determined as follows.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi mathvariant=¨normal¨»(«/mi»«mn»4«/mn»«msup»«mi»x«/mi»«mn»1«/mn»«/msup»«mi mathvariant=¨normal¨»)«/mi»«mo»+«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mi mathvariant=¨normal¨»(«/mi»«mn»1«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«mo»+«/mo»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«/mtd»«/mtr»«/mtable»«/math»
| The derivative of a constant is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math». |
Differentiate «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»-«/mo»«mn»3«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/math».
Use the distributive law to expand the function.
Apply the sum and difference rules, along with the power rule to find the derivative.
The derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo mathvariant=¨italic¨»`«/mo»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»3«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»15«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Apply the sum and difference rules, along with the power rule to find the derivative.
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»15«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo mathvariant=¨italic¨»`«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mn»2«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mn»15«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«mo»§#8722;«/mo»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mtd»«/mtr»«/mtable»«/math»
The derivative of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo mathvariant=¨italic¨»`«/mo»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«mo»+«/mo»«mn»2«/mn»«/math».