Calculus for Biological Scientists

Here we observe that neither \(4\) nor \(2a\) depend on the value of \(h\text{,}\) so as \(h \to 0\text{,}\) \((4 - 2a - h) \to (4 - 2a)\text{.}\) Thus, \(f'(a) = 4 - 2a\text{.}\)

This result is consistent with the specific values we found above: e.g., \(f'(3) = 4 - 2(3) = -2\text{.}\) And indeed, our work confirms that the value of \(a\) 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 \(a\text{,}\) \(x\text{,}\) 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 \(y = f(x)\text{,}\) as well as for the purpose of graphing the derivative function. For the function \(f(x) = 4x - x^2\text{,}\) it follows that \(f'(x) = 4 - 2x\text{.}\)

Because the value of the derivative function is linked to the graph of the original function, it makes sense to look at both of these functions plotted on the same domain.

In Figure 2.3.1, 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 graphs: 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 same as the height at the corresponding point on the righthand graph. That is, at each respective value of \(x\text{,}\) the slope of the tangent line to the original function is the same as the height of the derivative function. 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 \(f\text{.}\) On the righthand graph of \(y = f'(x)\text{,}\) the units on the vertical axis are units of \(f\) per unit of \(x\text{.}\)

An excellent way to explore how the graph of \(f(x)\) generates the graph of \(f'(x)\) is through an interactive like the one below:

In Section 2.2 when we first defined the derivative, we wrote the definition in terms of a value \(a\) to find \(f'(a)\text{.}\) 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.

Activity 2.3.2 included some graphs for which the associated derivative function was not defined for certain \(x\) values. The interactive below contains another example of a function \(f(x)\) for which \(f'(1)\) does not exist. Use the interactive to see how secant lines behave as the distance \(h\) from \(x=1\) gets closer to zero from both the left and the right. Compare this behavior to the interactive from Section 2.2, and write a thoughtful sentence explaining why it makes sense that \(f'(1)\) does not exist.

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 \(f'(a) = 4 - 2a\text{,}\) or equivalently that \(f'(x) = 4 - 2x\text{.}\) We subsequently graphed the functions \(f\) and \(f'\) as shown in Figure 2.3.1. Following Activity 2.3.2, we now understand that we could have constructed a fairly accurate graph of \(f'(x)\) without knowing a formula for either \(f\) or \(f'\text{.}\) At the same time, it is useful 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 \(f'(x)\text{:}\) given a formula for \(y = f(x)\text{,}\) the limit definition of the derivative will be used to develop a formula for \(f'(x)\text{.}\)