L3 The Power Rule and The Sum and Differences Rules of Derivatives - Part 1
Completion requirements
Unit 2A
Derivatives Part 1
Lesson 3: The Power Rule and The Sum and Differences Rules of Derivatives
Derivatives
When learning multiplication, you probably used physical representations like the one shown.
The illustrations show the essence of multiplication and are great for helping children understand what multiplication means. Unfortunately, drawing diagrams or using objects to compute multiplication problems can take a long time and are not practical when large numbers are involved. So, once multiplication is understood for small numbers, children work to develop more efficient strategies that could be applied to computations involving large numbers. These new strategies are great for saving time, but are useless if the children do not first understand what multiplication means.
In the previous Lesson, you determined the derivative of several functions using the definition of the derivative, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mo»(«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«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»«/math». This approach is like computing multiplication problems by drawing apples. It shows the essence of a derivative, but can be tedious to use and prone to errors, especially for complicated functions.
Much of the study of differential calculus involves learning alternative methods of finding derivatives. These methods, or rules, can help you save time. But, never lose sight of what a derivative actually is – the limit of a ratio.
The Power Rule
In Lesson 2, the derivatives of some of the following functions were determined using first principles. Some additional functions and their derivatives have also been included in the table. Do you see any patterns between each function and its derivative?
The power rule can be used to find the derivative of a function.
When learning multiplication, you probably used physical representations like the one shown.
The illustrations show the essence of multiplication and are great for helping children understand what multiplication means. Unfortunately, drawing diagrams or using objects to compute multiplication problems can take a long time and are not practical when large numbers are involved. So, once multiplication is understood for small numbers, children work to develop more efficient strategies that could be applied to computations involving large numbers. These new strategies are great for saving time, but are useless if the children do not first understand what multiplication means.
In the previous Lesson, you determined the derivative of several functions using the definition of the derivative, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mo»(«/mo»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«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»«/math». This approach is like computing multiplication problems by drawing apples. It shows the essence of a derivative, but can be tedious to use and prone to errors, especially for complicated functions.
Much of the study of differential calculus involves learning alternative methods of finding derivatives. These methods, or rules, can help you save time. But, never lose sight of what a derivative actually is – the limit of a ratio.
The Power Rule
In Lesson 2, the derivatives of some of the following functions were determined using first principles. Some additional functions and their derivatives have also been included in the table. Do you see any patterns between each function and its derivative?
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo»=«/mo»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/mrow»«/mstyle»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»2«/mn»«mi»x«/mi»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«/mrow»«/mstyle»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«/math» |
| «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»«/mrow»«/mstyle»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»4«/mn»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»f«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«/mrow»«/mstyle»«/math» | «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mo mathvariant=¨italic¨»`«/mo»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«mn»5«/mn»«msup»«mi»x«/mi»«mn»4«/mn»«/msup»«/math» |
The power rule can be used to find the derivative of a function.
| The Power Rule If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«/mrow»«/mstyle»«/math», then «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»y«/mi»«mo mathvariant=¨italic¨»`«/mo»«mo»=«/mo»«mi»n«/mi»«msup»«mi»x«/mi»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«/math».
Alternative notation: If «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mi»x«/mi»«mi»n«/mi»«/msup»«/mrow»«/mstyle»«/math», then «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»d«/mi»«mi»y«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»n«/mi»«msup»«mi»x«/mi»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«/mrow»«/mstyle»«/math».
The exponent «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»n«/mi»«/mstyle»«/math» of the original function appears as the coefficient of the derivative. The exponent «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math», appearing in the derivative, is one less than the exponent in the original function. |