The Derivative is Itself a Function
In your work in Preview Activity 1.4 with f(x)=4x−x2, you may have found several patterns. One comes from observing that f′(0)=4, f′(1)=2, f′(2)=0, and f′(3)=−2. That sequence of values leads us naturally to conjecture that f′(4)=−4 and f′(5)=−6. Even more than these individual numbers, if we consider the role of 0, 1, 2, and 3 in the process of computing the value of the derivative through the limit definition, we observe that the particular number has very little effect on our work. To see this more clearly, we compute f′(a), where a represents a number to be named later. Following the now standard process of using the limit definition of the derivative,
(f′(a)=lim
Here we observe that neither 4 nor 2a depend on the value of h, so as
h \to 0,\; (4-2a-h) \to (4-2a).
Thus, f'(a)=4-2a.
This observation is consistent with the specific values we found above: e.g., f'(3)=4-2(3)=-2. And indeed, our work with a confirms that while the particular value of a at which we evaluate the derivative affects the value of the derivative, that value has almost no bearing on the process of computing the derivative. We note further that the letter being used is immaterial: whether we call it a, x, or anything else, the derivative at a given value is simply given by “4 minus 2 times the value.” We choose to use x for consistency with the original function given by y=f(x), as well as for the purpose of graphing the derivative function, and thus we have found that for the function f(x)=4x-x^2, it follows that f'(x)=4-2x.
Because the value of the derivative function is so closely linked to the graphical behavior of the original function, it makes sense to look at both of these functions plotted on the same domain. In Figure 1.18, on the left we show a plot of f(x)=4x-x^2 together with a selection of tangent lines at the points we’ve considered above. On the right, we show a plot of f'(x)=4-2x with emphasis on the heights of the derivative graph at the same selection of points. Notice the connection between colors in the left and right graph: the green tangent line on the original graph is tied to the green point on the right graph in the following way: the slope of the tangent line at a point on the lefthand graph is the ![alt](https://math.libretexts.org/@api/deki/files/4380/clipboard_e7169fc12af15edbadbad5e5008697c51.png?revision=1)
Figure 1.18: The graphs of f(x)=4x-x^2 (at left) and f'(x)=4-2x (at right). Slopes on the graph of f correspond to heights on the graph of f'.
same as the height at the corresponding point on the righthand graph. That is, at each respective value of x, the slope of the tangent line to the original function at that x-value is the same as the height of the derivative function at that x-value. Do note, however, that the units on the vertical axes are different: in the left graph, the vertical units are simply the output units of f. On the righthand graph of y=f'(x), the units on the vertical axis are units of f per unit of x.
Of course, this relationship between the graph of a function y=f(x) and its derivative is a dynamic one. An excellent way to explore how the graph of f(x) generates the graph of f'(x) is through a java applet. See, for instance, the applets at http://gvsu.edu/s/5C or http://gvsu.edu/s/5D, via the sites of Austin and Renault5 .
In Section 1.3 when we first defined the derivative, we wrote the definition in terms of a value a to find f'(a). As we have seen above, the letter a is merely a placeholder, and it often makes more sense to use x instead. For the record, here we restate the definition of the derivative.
Definition 1.4
Let f be a function and x a value in the function’s domain. We define the derivative of f with respect to x at the value x, denoted f'(x), by the formula f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}, provided this limit exists.
We now may take two different perspectives on thinking about the derivative function: given a graph of y=f(x), how does this graph lead to the graph of the derivative function y=f'(x)? and given a formula for y=f(x), how does the limit definition of the derivative generate a formula for y=f'(x)? Both of these issues are explored in the following activities.
Exercise \PageIndex{1}
For each given graph of y=f(x), sketch an approximate graph of its derivative function, y=f'(x), on the axes immediately below. The scale of the grid for the graph of f is 1\times 1; assume the horizontal scale of the grid for the graph of f' is identical to that for f. If necessary, adjust and label the vertical scale on the axes for f'.
![alt](https://math.libretexts.org/@api/deki/files/3973/clipboard_ed3cef20e12041ad852624838164d9d83.png?revision=1&size=bestfit&width=561&height=762)
When you are finished with all 8 graphs, write several sentences that describe your overall process for sketching the graph of the derivative function, given the graph the original function. What are the values of the derivative function that you tend to identify first? What do you do thereafter? How do key traits of the graph of the derivative function exemplify properties of the graph of the original function? C For a dynamic investigation that allows you to experiment with graphing f' when given the graph of f, see http://gvsu.edu/s/8y. 6 6Marc Renault, Calculus Applets Using Geogebra. 39
Now, recall the opening example of this section: we began with the function y=f(x)=4x-x^2 and used the limit definition of the derivative to show that f'(a)=4-2a, or equivalently that f'(x)=4-2x. We subsequently graphed the functions f and f' as shown in Figure 1.18. Following Activity 1.10, we now understand that we could have constructed a fairly accurate graph of f'(x) without knowing a formula for either f or f'. At the same time, it is ideal to know a formula for the derivative function whenever it is possible to find one. In the next activity, we further explore the more algebraic approach to finding f'(x): given a formula for y=f(x), the limit definition of the derivative will be used to develop a formula for f'(x).
Activity \PageIndex{2}
For each of the listed functions, determine a formula for the derivative function. For the first two, determine the formula for the derivative by thinking about the nature of the given function and its slope at various points; do not use the limit definition. For the latter four, use the limit definition. Pay careful attention to the function names and independent variables. It is important to be comfortable with using letters other than f and x. For example, given a function p(z), we call its derivative p'(z).
- f(x)=1
- g(t)=t
- p(z)=z^2
- q(s)=s^3
- F(t)=\frac{1}{t}
- G(y)=\sqrt{y}