Unit 2A

Derivatives Part 1

Lesson 4: Product and Quotient Rules

The goals of the first three Lessons in Unit 2A were to develop a solid understanding of derivatives and to apply that understanding in an efficient manner to basic functions. In this Lesson, you will learn to find the derivatives of more complex functions using a combination of rules that will be developed, and then applied.

The Product Rule

The rectangle shown has a length of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»u«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» and a width of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»v«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math», where both «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»u«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»v«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mstyle»«/math» are positive differentiable functions.

What is the area of the rectangle?


Now, suppose the rectangle’s length is increased by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#9651;«/mo»«mi»u«/mi»«/mstyle»«/math» and its width is increased by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#9651;«/mo»«mi»v«/mi»«/mstyle»«/math». Determine the area of each of the three new rectangles formed?

Calculate the area of each additional rectangle in the diagram.