Calculus (III): What Is A Derivative? How To Really Integrate Differentials?

When it comes to mastering calculus, a sound understanding of the notion of a derivative is of paramount importance. It is surprisingly easy to establish a functional understanding of the derivative without deeply questioning the fundamentals.

However, this approach is likely to cause critical problems when one applies it to advanced calculus/analysis. This essay aims to eliminate such a pitfall.

This just happens to be the third in the sequence of the most fundamental calculus concepts that I have covered so far in the calculus series. In my previous two essays in this series, I covered the notion of a function and that of a limit.

In this essay, I will start by discussing the ground fundamentals that make up a derivative. Then, I will extend the discussion to what it means to actually integrate in the context of a derivative. I hope you are as excited as I am to dive into these topics. Without any further ado, let us begin.

This essay is supported by Generatebg

A Brief History of Time

Why don’t we begin by discussing the notion of time? We currently stack up 60 minutes into an hour, 24 hours into a day, and 7 days into a week; nothing fancy there. Let us now go back to the notion of minutes — have you ever wondered why they are called so?

You see, back in the day, people considered a minute so small in magnitude compared to an hour, that they named it a ‘minute’ (read the word like in “the minute (my-newt) dust particle”). At some point, even minutes did not provide enough resolution for certain applications.

So, the smart people of the day divided every minute into 60 further sub-divisions. These were called ‘second minutes’. In other words, these were small temporal quantities of the second order of smallness in comparison to minutes.

Over time, people shortened ‘second minutes’ to just ‘seconds’. And thus, seconds were born! I know what you must be thinking now:

“All this is very interesting. But where does the notion of a derivative come into all of this?”

Trust me, you will shortly find out that all this is very relevant to understand what a derivative is. For now, consider the following as the takeaway from this discussion:

In any measurement metric, there are some quantities that are of multiple orders of smallness (or bigness) compared to other quantities (like seconds-to-weeks or milligrams-to-tonnes).

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The Moving Car — Introduction to the Derivative

Let us now picture a car that moves at ten metres per second.

It starts from zero and reaches 100 metres after 10 seconds. Consequently, its displacement (y-axis) can be plotted against elapsed time (x-axis) as follows:

As you can clearly see, the displacement change is described by a straight line; time is the independent variable, while displacement is the dependent variable. This line, then, represents a linear function. In this case, we can describe the function using the following equation:

y = 10x

What this means is that at any given point in time, the displacement of the car is ten times the elapsed time. For example, after 7 seconds, the car would have traveled 70 metres. Even without calculus, we can tell from the graph that the instantaneous speed of the car is 10 metres/second.

However, notice the slope, which you can calculate by considering any two points on the straight line and then computing (y2 — y1)/(x2 — x1). You will notice that the slope at any point on this line is 10. This is also nothing special.

However, the slope also gives us the ‘derivative’ of this function. Expressed in formal terms, the derivative of the function (y = 10x) is simply the number 10. We will see what is going on here in a moment. But for now, just note that the derivative of this function tells us the following information:

1. At any point on this line, the speed of the car is 10 metres/second.

2. At any point on this line, its slope is 10.

Explanation: What is a Derivative?

The derivative of a function is merely ANOTHER FUNCTION that describes the rate at which the dependent variable changes with respect to the rate of change of the independent variable.

In simpler terms, the derivative tells us what happens to y when x changes. That is all it does!

For a linear function like the one we just saw, the dependent variable is varying at a constant rate with respect to the independent variable.

So, the derivative of such linear functions is simply the corresponding constant rate. In a more formal sense, we can express this as follows:

The derivative of (y = ax) is just the constant ‘a’.

That is simple, but what about the derivative of a constant? A constant does not vary with respect to the independent variable.

In other words, it is not dependent at all on the independent variable. In such a case, the derivative is simply zero. For example:

The derivative of y = 10 is just 0.

So, if the car came to a stop after 150 metres and stayed still, the derivative of the function describing its displacement (which would be a horizontal line) with respect to time (after it came to a stop) would just be zero.

You might think that all of this is trivial. And I would totally understand. Why would we need calculus to compute rates of change for linear functions and constants? Well, we need not force ourselves to use calculus for such cases.

However, calculus stops being trivial as soon as we start treating non-linear functions such as y=x².

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The Derivative of a Non-Linear Function

Let us now consider the function y = x². The cartesian graph of this function would look as follows:

Without going into the computational details for now, just take it from me that the derivative of this function is ‘2x’. What does this tell us about the rate of change of ‘y’ with respect to the rate of change of ‘x’?

To answer that question, it helps us to imagine ‘x’ as a square’s side length and ‘y’ as its area. In this case, the function’s derivative tells us that the area of the square is changing twice as fast as the rate of change of its side length.

Say that we start from x = 1 and proceed to x = 10 at the rate of 1 unit per second. When the square’s side length reaches the value of 10 units (after ten seconds), its area reaches the value of 100 units².

At this same point, the derivative of the function describing the area would tell us that the square’s area is increasing with respect to its side by 20 units (2*x = 2*10). Notice that the metric here is the side length.

Geometrically, the derivative would represent a tangent to the function graphing the curve at the said point.

The Derivative as a Limit

Let us now tabulate how the square’s side and area change with respect to time. The plot twist here is that I am going to focus on what happens between 3 and 4 seconds:

The average growth rate from time 3 to 3.1 seconds is:

(9.61–9)/(3.1–3.0) = 6.1 units² / second

The average growth rate from time 3 to 3.01 seconds is:

(9.0601–9)/(3.01–3.0) = 6.01 units² / second

The average growth rate from time 3 to 3.001 seconds is:

(9.006001–9)/(3.001–3.0) = 6.001 units² / second

Does this look familiar to you? It should, if you had read my last essay on the notion of limits. The average growth rate approaches 6 units²/second as the limit.

In other words, the derivative of the area with respect to the side or time (only in this case) is the limit of an infinite sequence of ratios. They just happen to converge on 6.

This is quite revealing. But let us move on and tackle the most common term in the context of a derivative.

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Conquering ‘dx’ and How to Really Integrate It?

Wehave been talking about the “derivative” thus far. But in mathematics, the term ‘dx’ is used ubiquitously in the context of a derivative. What does it really mean?

Well, ‘dx’ simply refers to an infinitely small fraction of x. This fraction is so small that it by itself can be considered negligible. This infinitely small fraction is formally known as a ‘differential’.

Now, you might be wondering why ‘dx’ is a negligible amount by itself. Well, we chose for it to have this property. We impose this requirement on it.

It is relatively easy to imagine that ‘x’ comprises an infinite number of dx’s. And if you sum up ALL of these dx’s, you get ‘x’. This is all that the operation ∫dx does.

In other words, the symbol ‘∫’ asks to you compute the sum of the relevant terms that follow. The term ‘integral’ merely refers to the ‘whole’ of the relevant terms that are being integrated.

Given that ‘dx’ itself is negligible, (dx*dx) is surely negligible as it represents a smaller quantity of second order of magnitude, just like how seconds were to weeks (from the beginning of the essay).

However, it DOES NOT follow that (x*dx) or (x²*dx) is necessarily negligible. This is because a very large number of infinitely small stuff added together might become relevant for computations.

The derivative of y with respect to x is expressed as dy/dx. In the past, dy/dx was known as the ‘differential coefficient’ (now defunct).

In a future essay, I will cover how dy/dx really refers to the derivative and how one can go about computing dy/dx (known as differentiating in the biz).

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References and credit: Silvanus Thompson and Martin Gardner.

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Further reading that might interest you:

• Is ‘0.99999…’ Really Equal To ‘1’?
• How To Benefit From Computer Science In Real Life (I)
• The Hindsight Bias: Cause And Effect

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