Calculus XII: How To Deal With The Time Derivative

Welcome to the twelfth entry in the calculus series, where we take a look at how to deal with the time derivative. The time derivative is one of the most fundamental and prominent applications of calculus. So, it is useful for any calculus beginner to spend some time understanding the time derivative properly.

In this essay, I will start by addressing some basic terminology associated with the time derivative. Then, I will go on to explain/illustrate some useful real-world applications of the time derivative.

At the end of this essay, you should be able to appreciate the usefulness of calculus in studying how real-world phenomena vary with time. Let us begin.

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Basic Terminology Associated with the Time Derivative

Let us start by understanding the term ‘rate’. In different contexts of the English language, and even in different contexts of mathematics, this term could mean different things. Often in mathematical contexts, ‘rate’ is used to describe a fraction or a proportion between two variables.

For example, traders use ‘rate’ to denote the price of commodities, like $25/piece (which means each piece costs $25). In the context of calculus, however, the term ‘rate’ takes on a more nuanced and specific meaning.

In calculus, the term ‘rate’ refers to how fast some variable varies with respect to time. For example, we call the rate at which a vehicle covers distance ‘speed’. In physics, when we denote both the direction and speed using one variable (a vector, mathematically speaking), we call it ‘velocity’.

By now, you might feel that ‘rate’ with respect to time is a simple concept. But let me linger a little longer on this topic to illustrate its conceptual depth.

Understanding ‘Rate’ Further

Suppose that a car is travelling at a constant rate (speed) of 20 metres per second (20 m/s). If we started measuring now, that car would have travelled 40 metres after 2 seconds. If we keep going, that car would have covered 1200 metres per minute. In other words, the car is travelling at a rate of 1200 metres per minute (1200 m/min).

But why stop there? Let us keep going. The same car is also travelling at a rate of 72000 metres per hour or 72 Kilometres per hour (72 Km/hr). By the same logic, the car is also travelling at 1728 Kilometres per day (1728 Km/day).

At this point, a typical person might start feeling uncomfortable. We know for sure that 20 metres are not the same as 1200 metres. Similarly, 72 Kilometres are not the same as 1728 Kilometres. Yet, how can all these rates be equal?

It is because this ‘rate’ is a fraction or a proportion of the distance covered per unit time. Note that time is the independent variable here, whereas the distance covered is the dependent variable.

As we vary our time measure (from seconds to minutes to hours to days), we get different distances in the numerator. But all these rates refer to the same fraction!

Each of these rates tells us how far the car would travel if it sustained this constant speed for the duration of the time in the respective denominator. All this is fine, but what does any of this have to do with calculus? Let us find out!

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Introducing the Time Derivative of Calculus

When you think about it, we are dealing with an independent variable (time) and a dependent variable (distance) here. Let us denote time by ‘t’ and distance by ‘x’.

Since the car is travelling at a constant rate, we can express the rate (speed) at which the car is travelling over an infinitesimally small time strip (‘dt’) as follows:

Rate of change of distance covered = dx/dt

And with just that trivial-looking conceptual shift, we have established that we can express rates with respect to time as derivatives! But the rabbit hole goes deeper.

You see, this rate is unlike the rate (speed) we saw before. This gives us the instantaneous rate of change of the car’s displacement with respect to time. To illustrate this subtle but non-trivial difference, let us look at another example.

Say that the same car travels from town A to town B in 4 hours. The distance between town A and town B is 200 Kilometres. This means that the rate at which the car travelled is 200/4 = 50 Kilometres/hour. But hang on a second! This is only true if the car travelled at a constant rate.

We know that the car started from rest in the beginning and slowed down to stop in the end. This means that the car must have increased its rate of travel in the beginning and decreased it towards the end.

So, the rate of 50 Kilometres/hour gives us the average rate. At any stage during the journey, the car could have travelled at a faster rate (like 60 Km/hr) or a slower rate (like 30 Km/hr) than the average rate.

Since the rate (dx/dt) applies over an infinitesimally small time strip, it gives us the instantaneous rate of the car’s travel. In other words, using this time derivative, we could measure instantaneous speeds during any moment of the car’s journey.

In calculus, the time derivative expresses the rate at which a dependent quantity changes with respect to time as the independent quantity. Let us now see further applications of the time derivative in physics.

Applications of the Calculus of the Time Derivative in Physics

In physics, the term ‘velocity’ is used to describe the time derivative along with a direction (a vector) as follows:

v = dx/dt

Now, let us go back to the beginning of the car’s journey from town A. When the car started from rest, it must have had an increased rate of displacement. How do we measure this rate of the rate of car’s displacement?

Well, that’s where the calculus concept of successive differentiation comes in handy; we can just differentiate velocity with respect to time:

The term ‘acceleration’, then, describes the rate of change of velocity with respect to time or the rate of change of the rate of change of distance with respect to time (with a unit of Km/hr/hr or Km/(hr)²).

When the car slows down, the acceleration would be negative, and in physics, we denote this as ‘deceleration’. We could keep going with higher order derivatives; the third time derivative of displacement is ‘Jerk’, the fourth derivative is ‘Snap’, the fifth ‘Crackle’, and the sixth ‘Pop’.

I have never needed to use anything above acceleration, though. Moving one step further, with Newton’s help, we know that we can express force as a product of mass and acceleration:

In physics, we can express the momentum of a body as a product of its mass and velocity as follows:

Momentum = m*v

If we differentiate Momentum with respect to time, we get the following result:

In other words, we have just established that force can also be expressed as the rate of change of momentum of a body. This example just goes to show us how useful calculus can be in understanding real-world phenomena that involve change with respect to time.

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Closing Comments — The Time Derivative of Calculus

The use of the time derivative is not only restricted to physics though. Another field where I have extensively used the time derivative is finance. Time derivatives can be used to quickly compute the rate at which an investment can vary.

The time derivative is especially useful for understanding behaviour of non-linear functions because human beings are typically bad at intuitively understanding non-linear rate of change.

In essence, the time derivative is not special compared to other derivatives in calculus. It just happens to find a vast array of useful applications in our day-to-day life. This is why I thought it made sense to write on this topic. I hope you enjoyed reading it.

Epilogue

In my essay on the thrilling story of calculus, I covered how Sir Isaac Newton and Gottfried Wilhelm Leibniz independently invented calculus. The notations of the time derivative that I demonstrated in this essay are actually from Leibniz.

Newton used different terminology; he described variables that continuously varied with time as ‘flowing’. Consequently, he termed the time derivative of such variables as ‘fluxions’. He then denoted fluxions using as ẋ, ẍ, etc.

Albeit being simple, this notation system has lost its place in calculus over the years because it does not communicate what the independent variable is.

However, I still come across this notation system in physics, when time is the independent variable. So, as far as I know, this notation system only holds for the time derivative — another unique property of the time derivative.

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Reference and credit: Silvanus Thompson.

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Further reading that might interest you:

• Is Zero Really Even Or Odd?
• Modern Math Is Full Of Symbols. Is This Really Necessary?
• Non-Euclidean Geometry: The Forgotten Story

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