L1 Higher Order Derivatives - Part 1
Completion requirements
Unit 2B
Derivatives Part 2
Lesson 1: Higher Order Derivatives
The derivative of function «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» is also a function, «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mi mathvariant=¨normal¨»`«/mi»«mo»=«/mo»«mi»f«/mi»«mi mathvariant=¨normal¨»`«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math». As such, you can take the derivative
of the derivative, known as the second derivative, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mi»``«/mi»«mo»=«/mo»«mi»f«/mi»«mi»``«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math».
Given «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math», predict «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mi»``«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math». What about «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»g«/mi»«mo»```«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math»? How many unique higher derivatives will «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» have?
The slope of the line tangent to a curve, also known as the rate of change, is found from the first derivative. Therefore, the second derivative gives the rate of change of the slope of the tangent line. In Unit 7, more investigation into the second derivative will show the connection between displacement, velocity, and acceleration.
Given «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»=«/mo»«msup»«mi»x«/mi»«mn»3«/mn»«/msup»«mo»§#8722;«/mo»«mn»3«/mn»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»4«/mn»«mi»x«/mi»«mo»§#8722;«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math», predict «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mi»``«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math». What about «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»g«/mi»«mo»```«/mo»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math»? How many unique higher derivatives will «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»g«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» have?
The slope of the line tangent to a curve, also known as the rate of change, is found from the first derivative. Therefore, the second derivative gives the rate of change of the slope of the tangent line. In Unit 7, more investigation into the second derivative will show the connection between displacement, velocity, and acceleration.