$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

2: Instantaneous Rate of Change: The Derivative

• • Contributed by David Guichard
• Professor (Mathematics) at Whitman College

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

• 2.1: The Slope of a Function
Suppose that y is a function of x, say y=f(x). It is often necessary to know how sensitive the value of y is to small changes in x.
• 2.2: An Example
We started the last section by saying, "It is often necessary to know how sensitive the value of y is to small changes in x .'' We have seen one purely mathematical example of this: finding the "steepness'' of a curve at a point is precisely this problem. Here is a more applied example.
• 2.3: Limits
In the previous two sections we computed some quantities of interest (slope, velocity) by seeing that some expression "goes to'' or "approaches'' or "gets really close to'' a particular value. In the examples we saw, this idea may have been clear enough, but it is too fuzzy to rely on in more difficult circumstances. In this section we will see how to make the idea more precise.
• 2.4: The Derivative Function
We have seen how to create, or derive, a new function f′(x) from a function f(x), and that this new function carries important information. To make good use of the information provided by f′(x) we need to be able to compute it for a variety of such functions.
• 2.5: Adjectives for Functions
It is useful to introduce a collection of adjectives to describe certain kinds of functions; these adjectives name useful properties that functions may have. It would clearly be useful to have words to help us describe the distinct features of the behavior of different functions. We will point out and define a few adjectives (there are many more) for the functions pictured here.
• 2.E: Instantaneous Rate of Change: The Derivative (Exercises)
These are homework exercises to accompany David Guichard's "General Calculus" Textmap.