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2: Derivatives

Last updated

Dec 21, 2020
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- 1.E: Applications of Limits (Exercises)
- 2.1: Instantaneous Rates of Change- The Derivative

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The previous chapter introduced the most fundamental of calculus topics: the limit. This chapter introduces the second most fundamental of calculus topics: the derivative. Limits describe where a function is going; derivatives describe how fast the function is going.

Topic hierarchy

2.1: Instantaneous Rates of Change- The Derivative
This section defined the derivative; in some sense, it answers the question of "What is the derivative?''
2.2: Interpretations of the Derivative
This section falls in between the "What is the definition of the derivative?'' and "How do I compute the derivative?'' sections. Here we are concerned with "What does the derivative mean?'', or perhaps, when read with the right emphasis, "What is the derivative?'' We offer two interconnected interpretations of the derivative, hopefully explaining why we care about it and why it is worthy of study.
2.3: Basic Differentiation Rules
In this section (and in some sections to follow) we will learn some of what mathematicians have already discovered about the derivatives of certain functions and how derivatives interact with arithmetic operations. We start with a theorem.
2.4: The Product and Quotient Rules
The previous section showed that, in some ways, derivatives behave nicely. However, the derivatives of other functions are not as straightforward for these, we need the Product and Quotient Rules, respectively, which are defined in this section.
2.5: The Chain Rule
To complete the list of differentiation rules, we look at the last way two (or more) functions can be combined: the process of composition (i.e. one function "inside'' another). The derivative of such compositions functions employs the new rule this section introduces, the Chain Rule.
2.6: Implicit Differentiation
Sometimes the relationship between y and x is not explicit; rather, it is implicit. Sometimes the implicit relationship between x and y is complicated. In this case there is absolutely no way to solve for y in terms of elementary functions. The surprising thing is, however, that we can still find y′ via a process known as implicit differentiation.
2.7: Derivatives of Inverse Functions
2.E: Applications of Derivatives(Exercises)

Contributors and Attributions

Gregory Hartman (Virginia Military Institute). Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License. http://www.apexcalculus.com/

Contributors and Attributions

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