DERIVATIVES OF A FUNCTION


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.

DEFINITION OF DIFFERENTIABILITY ON A SET
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 0 (revisited). The constant function f(x) = 2 is differentiable (on its entire domain R).

Example 1 (revisited). The linear function f(x) = 2x is differentiable (on its entire domain R).

Example 2 (revisited). The quadratic function f(x) = x2 is differentiable (on its entire domain R).

Example 3 (revisited). The square root function f(x) = sqrt(x) is differentiable on the open ray ( 0 , infinity ).

THE DERIVATIVE AS A FUNCTION
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.

Limit computations from the Examples on the previous page reveal the following derivative formulas.

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) = x2 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) = u2, 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) = x2, we write f '(x) = 2x.

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OK, so that's the derivative. Here are three issues of central concern.

  1. Interpretation: What does the number f '(a) mean?
  2. Computation: How does one compute f '(a)?
  3. Application: What use can be made of the derivative?
The remainder of Stage 5 is devoted to the issue of Interpretation. The principal interpretations concern instantaneous rate of change and tangency. Not surprisingly, the question of interpretation is intimately linked to the issue of application. The interpretation of the derivative as rate of change is largely responsible for the applicability of differential calculus to real life problems. We will study many applications in later Stages.

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.

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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) = x2. The first three derivatives of f are the following.

We leave it to you to verify that all remaining derivatives of f are also constant at 0: For each real number x, f(4)(x) = f(5)(x) = ... = 0.


Equipment Check.

Here are some opportunities to check your understanding of the basic concepts, terminology, and notation.
Differentiation of the second derivative of f produces...
the derivative of f.
f '''.
the rate of change of f.
Check Answer Feedback here

Please consider the function f(x) = 3x2 - 2 and the following limit.

(3(7 + h)2 - 2) - (3(7)2 - 2)
lim
h-->0h
Which of the following is equal to the limit displayed above?
f '(a)
f '(h)
f '(7)
Check Answer Feedback here

Still with f(x) = 3x2 - 2, please construct a suitable limit problem and compute f '(-1). Your answer should be a real number.

Compute f '(-1)
Enter answer here
Check Answer Feedback here
Explain

Having met the basic terminology of the derivative, we now discuss the issue of notation.


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