In many cases, a function is differentiable at many (or even all) points in its domain; here is some convenient terminology to describe this situation. Let I be a subset of the domain of a function f.
The function f is differentiable on I if f is differentiable at a for each point a in I. The function f is differentiable if it is differentiable at each point of its domain. |
Example 1 (revisited). The linear function f(x) = 2x is differentiable (on its entire domain R).
Example 2 (revisited). The quadratic function f(x) = x^{2} is differentiable (on its entire domain R).
Example 3 (revisited). The square root function f(x) = sqrt(x) is differentiable on the open ray
If a given function f is differentiable at a domain value a, then f '(a) is a real number. In this way f ' itself becomes a new function, called the derivative of f. The domain of the derivative is the set of all points in the domain of f at which f itself is differentiable. |
Example 0 (revisited). If f(x) = 2 for all x in R, then f '(a) = 0 for all a in R.
Example 1 (revisited). If f(x) = 2x for all x in R, then f '(a) = 2 for all a in R.
Example 2 (revisited). If f(x) = x^{2} for all x in R, then f '(a) = 2a for all a in R.
Example 3 (revisited). If f(x) = sqrt(x) for all x nonnegative real numbers x, then f '(a) = 1/2sqrt(a) for all positive real numbers a. Even though f(0) is defined (and is equal to 0), the value f '(0) is not defined when f is the square root function.
Of course, we are free to use other letters to denote functions and their independent variables. For instance, if g(u) = u^{2}, then g'(u) = 2u. If h(z) = sqrt(z), then h'(z) = 1/2sqrt(z). In fact, we will often use x as the independent variable for f ', even when we have already used it for f. Thus, given the function f(x) = x^{2}, we write f '(x) = 2x.
OK, so that's the derivative. Here are three issues of central concern.
Stage 6 is devoted to the issue of computation: Given f and a, how do we compute f '(a)? The first answer to this question is that one computes a suitable limit of difference quotients. That is an extremely laborious process! Happily, there is an easier way; we will study a variety of techniques that make the process of computing derivatives, called differentiation, much simpler.
Before moving on to these topics, let us first define the higher derivatives of a function. We start with a function f whose domain and target set consist of real numbers. The derivative f ' is a new function whose domain consists of all points at which f is differentiable and whose target set consists of real numbers. We can differentiate f ' in turn to obtain its derivative. This is the second derivative of f, and it is denoted f ''. (What else!!?!) This process can be repeated, and the result is a progression that leads to the third, fourth, fifth derivatives of f, and so on. We eventually abondon the "prime" notation f ', f '', f ''' in favor of the notation f^{(4)}, f^{(5)},... for fourth and higher derivatives. The questions of interpretation, application, and computation are all relavent for these new objects as well.
Examples 2, 1, and 0 (revisited). Let f(x) = x^{2}. The first three derivatives of f are the following.
Please consider the function f(x) = 3x^{2} - 2 and the following limit.
(3(7 + h)^{2} - 2) - (3(7)^{2} - 2) | |
lim | |
h-->0 | h |
Still with f(x) = 3x^{2} - 2, please construct a suitable limit problem and compute f '(-1). Your answer should be a real number.
Having met the basic terminology of the derivative, we now discuss the issue of notation.
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© CalculusQuest^{TM}
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson