Overview
Having now completed Chapter 1 on the definition and meaning of the derivative, we turn to Chapter 2 where our principal focus will be on computing derivatives. By this we mean that we want to further understand the limit definition of the derivative and patterns that can be found in certain classes of functions, patterns that will enable us to simply look at the formula for a function and then be able to write down a formula for the derivative. As we progress through Chapter 2, you’ll see that we will build from the simplest functions to much more complicated ones.
This section covers the following concepts: Notations for derivatives. The Power Rule. Differentiation of exponential functions. The derivative of constant multiples, sums, and differences of functions.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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Adequately use the Leibniz notation for derivatives: Explain what the notation \(\displaystyle \frac{dy}{dx}\) means as well as the notation \(\displaystyle \frac{d}{dx}\).
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Use elementary differentiation rules to differentiate polynomials, power functions, and exponential functions.
- Use the constant rule and the sum and difference rule to differentiate combinations of the above functions.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
- Use the rules introduced in this section to differentiate functions that are not given as a formula.
To prepare for class
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Read the beginning of section 2.1 in Active Calculus, up to and including subsection 2.1.1 “Some Key Notation”.
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Do the Preview Activity for this section (on WeBWorK, if required by your teacher).
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Watch the following video which summarizes the different differentiation rules that we will see in this section.
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Watch the following video which explains how to simplify functions before being able to apply the basic differentiation rules (video created at CCSL):
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If you feel that more explanations are needed to understand these rules, read subsections 2.1.2 (“Constant, Power, and Exponential Functions”) and 2.1.3 (“Constant Multiples and Sums of Functions”).
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Watch the following videos which explain how these rules are used (except for the sum rule which will be dealt with in class).
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Think about where some of the rules for differentiation (Constant Multiple Rule, Sum Rule) actually come from. Don’t they look familiar?
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Do some experimentation with the following interactive applets:
After class
- Finish any in-class activities you might not have finished during class.
- Do the problems on the WeBWorK assignment for this section.
