A.5 Differentials; Begin to Determine 𝑑𝑓𝑑𝑥 at 𝒙 =𝒂

On this screen we're going to introduce differentials, a key Calculus concept, by building from the ideas you used in your simple calculations on the preceding screens. We'll also use those ideas to lay the groundwork for how to determine the rate at which a function changes at a given point.

Summarizing What We've Done So Far

In the preceding Topic, we developed the method of linear approximations to compute a variety of values for a few different functions. For each calculation, the problem statement provided

• the function itself. For instance in Example 1, 𝑓(𝑥) =𝑥2.
• a particular value of 𝑥 for which we can easily compute the function. For instance, at 𝑥 =3, 𝑥2 =9.
• the rate at which the function changes at this particular value of 𝑥. For instance, 𝑑𝑓𝑑𝑥∣at 𝑥=3 =6.

The table below shows each of the calculations we considered, starting with Example 1 in row 1, where 𝑓(𝑥) =𝑥2.

Notice that in the second column we're using the letter a to represent the input (x or 𝜃 value) that we used as our "base point" from which we started our approximation. For instance, in row 1 we have 𝑎 =3 and 𝑓(𝑎) =32 =9.

With that overview of calculations in mind, let's summarize some key points about what we've done so far. While we're using the particular computations we've completed to illustrate the following primary concepts, keep in mind that these ideas will apply to most functions we will encounter.

1. In each problem, we focus on a point at 𝑥 =𝑎 (column 2) for which we know both (I) the function's value at that point (column 3), and (II) the rate at which the function changes at that point, 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎 (column 4).
   So far we've had to simply provide that rate for you. That's about to change.
2. Using our linear approximation method we can calculate approximate values of the function for values of x that are a small distance 𝑑𝑥 away from 𝑥 =𝑎. For instance, in Example 1 we had 𝑑𝑥 =0.01.
3. We can envision our linear approximation method as starting at the point we know about (𝑎,𝑓(𝑎)), and then walking along the line with slope equal to 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎 instead of following the function's actual curve.
4. Walking along this line, the small change in the function's output value 𝑑𝑓 is directly proportional to the small change in x-value 𝑑𝑥. The constant of proportionality is the function's rate of change 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎: 𝑑𝑓=rate of change at 𝑥=𝑎⏞¯¯⏞¯¯⏞(𝑑𝑓𝑑𝑥∣at 𝑥=𝑎)⋅𝑑𝑥
5. The function's rate of change 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎 is thus a measure of the function's sensitivity at 𝑥 =𝑎 : the rate of change at that point determines how strongly the function reacts when you change its input by the small amount dx. For example, in the the interactive figure below, at (a) 𝑥 =𝑎1 the function's rate of change 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎1 is larger than at (b) 𝑥 =𝑎2. Using the slider beneath the graph you can vary the size of dx, which causes the identical changes in the horizontal direction in both (a) and (b). Observe that because 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎 is larger at 𝑎1 than at 𝑎2, the function "reacts more strongly" at 𝑎1, and so the change dy is much larger at 𝑎1 than at 𝑎2 for the same change in dx.

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Check Question

The interactive graph below shows the function g(x) versus x. You can use the slider beneath the graph to vary the size of dx near each point.

Choose the correct statement that orders the rates of change at the four points from most negative to most positive.

(a) 𝑑𝑔𝑑𝑥∣at 𝑥=𝑎1 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎2 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎3 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎4

(b) 𝑑𝑔𝑑𝑥∣at 𝑥=𝑎2 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎4 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎3 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎1

(c) 𝑑𝑔𝑑𝑥∣at 𝑥=𝑎2 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎1 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎4 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎3

(d) 𝑑𝑔𝑑𝑥∣at 𝑥=𝑎1 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎3 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎4 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎2

(e) 𝑑𝑔𝑑𝑥∣at 𝑥=𝑎4 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎3 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎2 <𝑑𝑔𝑑𝑥∣at 𝑥=𝑎1

✓ ✘ A ✓ ✘ B ✓ ✘ C ✓ ✘ D ✓ ✘ E

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To answer this question, first choose a positive value for dx, as shown in the figure. (You know dx is positive if its horizontal line extends to the right from the red dot that marks the point 𝑥 =𝑎.) You may have chosen a different value for dx than we did; it doesn't matter, as long as you can easily see the different changes in dg that result.

Then look first at the direction (up or down) of each change dg: in (1) and (3), dg is negative. Furthermore, the most negative change happens at (1): the negative change there is larger than the one in (3). Hence the first item in our ordered list will be 𝑑𝑔𝑑𝑥∣at 𝑥=𝑎1. We already know, then, that the correct answer choice will be (a) or (d).

Next, notice that the most positive change happens at (2). Hence the last item in our list must be 𝑑𝑔𝑑𝑥∣at 𝑥=𝑎2. We thus know that the correct answer choice is (d).

But let's continue our reasoning anyway: At (3), dg is negative, but less so than at (1) since the size of its dg is smaller than at (1). And at (4) dg is positive, but less so than at (2). Hence our complete ordering is

most negative <smaller negative <smaller positive <most positive𝑑𝑔𝑑𝑥∣at 𝑥=𝑎1<𝑑𝑔𝑑𝑥∣at 𝑥=𝑎3<𝑑𝑔𝑑𝑥∣at 𝑥=𝑎4<𝑑𝑔𝑑𝑥∣at 𝑥=𝑎2⟹(d)✓

Linear approximation means direct proportionality between small change in input and small change in output

You've probably noticed that in each of the graphs above, and in the graph for each linear approximation calculation we've done, you see a triangle. This is an important realization! The triangle comes about by definition of "linear approximation": in this approximation method, each function's small change in output-value df (or dg or d-whatever-output) near the "base point" is always directly proportional to the small change in input dx (or 𝑑𝜃 or dt or d-whatever-input):

Defining differentials

With that fundamental idea in mind, let's introduce some new terminology: The small-change quantities df, dg, ds, dx, 𝑑𝜃, and dt are all called differentials. As we've seen,

• differentials are represented by placing a d in front of the variable;
• differentials represent small changes in the variable's value;
• a function's output-differential df (or dg or ...) changes at a constant rate with respect to the function's input-variable's differential dx (or 𝑑𝜃 or ...).

Leibniz notation and Leibniz triangles

Differentials were created by Gottfried Leibniz (1646-1716), which is why you might hear quantities like 𝑑𝑓𝑑𝑥 referred to as Leibniz notation. And the triangles we've been forming, with the small change in input (dx, say) as the base and the small change in output (df, say) as the height, are known as Leibniz triangles. [Link to Wikipedia article about Leibniz.]

Flipping the script: Using differentials to determine 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎

So let's shift our focus to this question, starting with a reversal of the problem-type we've been considering: in each problem we've examined so far, we've provided the function 𝑓(𝑥), a point of interest, 𝑥 =𝑎, and the value of 𝑓(𝑎) and the rate at which the function changes at that value of x. We've then essentially asked you to compute the value of df using 𝑑𝑓=(𝑑𝑓𝑑𝑥∣at 𝑥=𝑎)⋅𝑑𝑥 For instance, the first approximation we did was for 𝑓(𝑥) =𝑥2 near 𝑥 =3, so 𝑓(3) =9. We then provided the rate 𝑑𝑓𝑑𝑥∣at 𝑥=3 =6 and asked you to compute the value of df for 𝑑𝑥 =0.01 so we could estimate 𝑓(3.01), which as you might recall, is (3.01)2 ≈9.06.

Let's switch things up, and provide you with values for df and dx and then ask you to determine the rate 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎. This is our first step toward finding multiple ways to determine the rate at which a function changes. Said differently, we looking to determine the function's exact "sensitivity" at the the point of interest.

To illustrate the idea, let's return to that very problem of what happens when we vary the function 𝑓(𝑥) =𝑥2 a bit, around the point 𝑥 =3. For the purposes of this Activity, pretend that you do not know already know that 𝑑𝑓𝑑𝑥∣at 𝑥=3 =6, and instead must determine that value.

Time for you to try a problem, this time with a result you don't already know.

The Upshot

1. Differentials are small changes in a variable's value. For instance, dx is a small change in input value, and df is the resulting small change in the output value.
2. By definition, df and dx are related according to 𝑑𝑓 =(𝑑𝑓𝑑𝑥∣at 𝑥=𝑎) ⋅𝑑𝑥, where 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎 is the rate at which the particular function changes at 𝑥 =𝑎, and is thus a measure of the function's "sensitivity" to changes in input-value at that point.
3. When we write this rate as 𝑑𝑓𝑑𝑥, we are using Leibniz notation, named after the person who invented it. The triangle that is formed with dx as its base and df as its height is known as a Leibniz triangle.
4. While on the preceding screens we provided the value of 𝑑𝑓𝑑𝑥∣at 𝑥=𝑎, we are laying the groundwork for how to determine it, which will lead to one of the Big Ideas in Calculus.

For now, if you have a question or thought about what's on this screen, please pop over to the Forum and post it there!