Coordinated Calculus

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

The left half of Figure 1.61 above shows a plot of \(y=f(x)\text{,}\) where \(f(x) = 4x - x^2\text{,}\) together with a selection of tangent lines at the points we considered in Example 1.60. The right half of the figure shows a plot of \(y=f'(x)\text{,}\) where \(f'(x) = 4 - 2x\text{,}\) with emphasis on the \(y\)-coordinates 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 left-hand graph is the same as the \(y\)-value at the corresponding point on the right-hand 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 output 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 a java applet. See, for instance, the applets at http://gvsu.edu/s/5C

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gvsu.edu/s/5C

or http://gvsu.edu/s/5D

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gvsu.edu/s/5D

, via the sites of Austin and Renault.

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David Austin, http://gvsu.edu/s/5r; Marc Renault, http://gvsu.edu/s/5p.

When we first defined the derivative in Section 1.4, 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.

For a dynamic investigation that allows you to experiment with graphing \(f'\) when given the graph of \(f\text{,}\) see http://gvsu.edu/s/8y

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gvsu.edu/s/8y

.

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Marc Renault, Calculus Applets Using Geogebra.

Now, recall the opening example of this section: we began with the function \(f(x) = 4x - x^2\) 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 \(f'\) in Figure 1.61. Following Example 1.62, we now understand that we could have constructed a fairly accurate graph of \(f'(x)\) without knowing a formula for either \(f\) or \(f'\text{.}\) Even so, it is useful to know a formula for the derivative function whenever it is possible to find one.

In the next example, we further explore the more algebraic approach to finding a derivative: given a formula for \(y = f(x)\text{,}\) the limit definition of the derivative will be used to develop a formula for \(f'(x)\text{.}\)