Derivative as a "rail" for growth

As I have another Blog in Spanish where I mix Mathematics with law, literature and no matter what, I am going to be more restrained here and speak only about Math and Physics. 

Since I wrote latest posts, I have learnt much more, so I think that I can aim at posting only things that should be orthodox, even if I cannot help being somehow “creative”.

Today’s post is about a visual way for understanding a differentiation rule, the “power rule”.

Let us choose a well-known function, like L = f(t), which takes as input the time elapsed and generates as output the length traversed by a body within that time lapse.

This function, like any function, can be viewed as the product between the input and the average change rate. This is self-evident, since L is obviously v * t = (L/t) * t = L.

In the simplest manifestation of the function (say for example L = 2t), v is a constant, so it will be the same no matter the interval. If the body has traversed 2 meters, it will have done it in 1 second, so the ratio 2/1 is 2.  If displacement is 1 meter, time required will be 0.5 seconds, the ratio being again 2.  And what if we keep reducing the interval considered until it is 0, what we call an “instant” without duration? Well, division by 0 is not permitted: it leads to an “undefined”, i.e. an expression that breaches the rules about how to build mathematical expressions… This shouldn’t come as a surprise: if you define velocity as a rate of change over time, how come that you now want to calculate it without allowing for the lapse of any time at all? However, here we can perfectly stipulate that this playful instantaneous rate is the same as in all other cases, since it is in fact constant: it is 2 as well.

In a more complicated form, the function is L = t². In our example, this will happen when the body is subject to acceleration.  Let me remind you that the kinematic formula for obtaining the displacement of a body that starts with velocity “v_o” and suffers acceleration “a” is L = v_o + at²/2. If we stipulate that the body starts at rest (v_o =0) and is subject to a = 2 m/s², then we get the above mentioned function, L = t². This can still be viewed as L = tt, where t is again the average velocity, but not the instantaneous one anymore, as velocity is continuously changing (there is acceleration). (It may surprise you that I refer to t as a velocity, but it is fine, because I mean its numerical value; if 3 seconds have elapsed and the body has displaced 3*3=9 meters, what I mean is that it has moved at an average of v m/s during t seconds, it just happens that the numerical value of v is the same as the number of seconds t having elapsed).

Because this instantaneous velocity is variable, it will be a function (the derivative or L’), where the argument can only be the one that we are in possession of (time), although arranged differently. For this sort of functions, the so-called “power functions” [where the argument is raised to progressively higher powers (t, t², t³…)], the rule for finding the derivative is the so-called “power rule”, which is L’ = n * t^n-1, i.e. (i) reduce the power by 1 and (ii) multiply by the power.

We can learn this rule by heart or try to understand the logic behind it. 

I asked about such logic here, but I am not very convinced with the answer, which extracts the proof from another differentiation rule, the product rule. My intention is however finding a logic that could precisely later illuminate such other rule.

Another approach is doing the normal operation that you carry out to find derivatives and then repeat it with a few power functions, until a pattern emerges.

In particular, with the function L = t², you would reason as follows:

If you repeat this operation with L = t³, the result is L’ = 3t² and thus the pattern represented by the power rule (L’ = n * t^n-1) clearly shines up.

I am not very convinced, however, about this idea that the interval “tends to zero in the limit”, because this suggests that the value obtained for the derivative is an “approximation”. But it is not such thing, it is an exact value. It is true that in the real world you can only measure a velocity as a ratio between distance and time, but the abstract idea of a body having a state of motion at a given “instant” (null interval) is perfectly valid. Precisely the visual interpretation that I propose is a description of things where you can dispense with the concept of a ratio. In particular, the idea is viewing the function (L = length traversed in our original example) as the area (A) (or volume or hyper-volume) of a geometric figure, which is stretched as you pull in the direction of the input from one or several sides, which side/s is/are thus providing a sort of path or rail constraining the growth of the figure.

The simplest manifestation is what we can call rectangular growth, as shown in these pictures:

In both cases, the function is the same: it is A = 3 x (which is the equivalent of L = 3 t in the original example).  The characteristic of this function is that there is a neat separation of roles: one dimension (either the horizontal or the vertical one, the choice is arbitrary) plays the role of input or growing dimension that pulls from the other side and is logically variable; you take this out and what is left, the other dimension, is fixed and one can view it as the side that you pull from to stretch the figure, which is acting as a path or rail constraining the growth thereof.

Next step is what we can call triangular growth, like in these pictures:

Again the function is the same for both pictures: it is A = x²/2 (which would be L = t²/2 in the original). Here both dimensions are growing harmoniously at the same time. After taking out one dimension, what is left is average growth rate, which is x/2, a half-side.  In turn, the derivative will be the side that you pull from and is therefore determining how much the area grows, which is one side (x). The choice of which side to pull from is arbitrary: in the first picture I chose to pull from the vertical one; in the second one we pull from the horizontal side; but in any case we pull from only one side. The novelty now is that, yes, certainly, such path is not fixed, it is variable, but that is not a problem. We said that the derivative would be the side from which we pull and that constitutes the path for growth of the other dimension and there is no reason for changing our mind just because such path is variable: it is in fact continuously widening following the hypotenuse of the triangle and that is why we say that the derivative, instead of the coefficient 3 or whatever number, is a variable, x_v as in the left picture or x_h as in the right one (any of them is fine, since both grow harmoniously).

In turn, a square-like growth, represented by the function A = x², looks as follows:

The average growth rate is one side (x), which is what is left after taking out one dimension. What is the derivative? It is two sides of the square, that is to say, 2x. The visual reason is that in order to obtain a square-like growth, as suggested by the expression x², you must pull from two sides, because (unlike what happened in the triangle case) only this way can you achieve to cover the area that the expression x² demands. We could expect this by realizing that the square area x² is the sum of the areas of two triangles (x²/2 + x²/2), so the growth path should also be double (x_h + x_v = 2x). And, yes, certainly, these paths are both variable, but that is not a problem: it just happens that each side is acting as a rail for the growth of the other one and growing itself within the other’s channel. So, in this case, we have two weird aspects: unlike in the rectangular case, the path is variable and, unlike in the triangle case, there is a mixture of roles, as both sides are reciprocally and simultaneously pulling from and constraining each other. But that is not dramatic, we just address the situation mutatis mutandi: because the path is variable, the derivative is x; because there are two paths, it is 2x.

If we now face a cube-like growth, where the function is V = x³, we can infer that the average growth rate will be one side, a face of the cube (x²) and the derivative, based on the visual criterion, should be 3 sides (3 faces of x² each = 3x²). Why? Because that is what you pull from to make the cube grow in its three dimensions and also grow along the full volume embraced by those sides, as demanded by the expression of the function.  The sides will be double than the dimensions, but you just need to pull from the dimensions (half of the sides), what we could call a “half-skin”.

From here onwards, with 4D or higher-dimensional figures (so called “hyper-cubes”), you apply the same algebraical rules:

•          since you must pull from the sides, to find yourself at the level thereof, you go down one dimension (you reduce the power by one) and
•          since you must pull from as many sides as dimensions has the figure, you multiply by the number of dimensions (which is the power itself).

Of course, as the case of the triangle exemplifies, all this is assuming that there is growth to the full extent of the figure (and not more); if you want a partial (or an extended) growth, the function will include the corresponding coefficient and so will the derivative.

Conclusion: the objective was finding a way to describe the derivative (instantaneous rate of change), at least for power functions, that is not relying on a ratio and I think that we have found it in this idea that the derivative is the side/s that you pull from in order to stretch a geometric figure.

Some other aspects to be discussed to complete the study:

•          What if the input of the function is not a side of the figure but its radius?
•          Can you see the function as a sum of individual growths in each dimension and the derivative as the sum of the velocities in each dimension?
•          Link with the calculation of initial / final velocity in kinematic formulas.
•          How this approach can serve also to explain product rule and other differentiation rules.
•          The link of all this with Zeno paradoxes.