Calculus XI: How To Deal With Successive Differentiation?

Welcome to the eleventh entry in the calculus series, where we see how to deal with successive differentiation. Although the term “successive” sounds intimidating, the mathematical concept behind is pretty straightforward.

To understand how it works, I will start by illustrating a practical example first and then proceed to cover the theoretical principles behind it. Following this, I will be paying special attention to the notation used in successive differentiation; people often take this for granted.

Finally, I will briefly explore the applications of successive differentiation in the scientific world. Without any further ado, let us begin.

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Successive Differentiation — A Practical Example

Consider the following function:

y = f(x) = 5x⁴

Let us say that we are interested in differentiating this function successively as many times as we can. We could do this with the help of the power rule that I have previously covered in the series.

As you can see, we managed to differentiate the function five times successively before we ended up with zero. The reason why we obtained a zero is that the constant (120) does not vary with respect to ‘x’ at all.

That aside, all I did here was differentiate the functions that resulted from differentiating the original function successively. This is all successive differentiation is. Pretty simple, right? However, the next point is not so trivial.

If you were curious, you would have noted that I represented the first differentiation using ‘dy/dx’, but I just wrote “??” for the rest of the successive operations. How can we represent these operations and what exactly do they mean?

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Notations for Successive Differentiation

Let us first consider the second successive differentiation, where we differentiate the derivative (dy/dx) with respect to ‘x’. This operation can be represented as follows:

It is important to note that the ‘2’ you see in the above expression is purely a notation, and does not represent a mathematical exponent in the strictest sense.

Gottfried Wilhelm Leibniz, one of the inventors of calculus, manipulated these symbols in a clever way such that they could be treated as operators. The ‘d²’ in the numerator by itself does not take the conventional mathematical definition, since ‘d’ is neither a variable nor a constant; it is an operator.

Furthermore, in the past, mathematicians wrote the denominator as (dx)², but nowadays, ‘dx²’ has become the norm. If we were to extend this logic to the example we solved above, we would arrive at the following results:

If we generalise, the successive nth differentiation/derivative of a function ‘y’ of ‘x’ can be written as follows:

Although practical, writing all of these symbols is cumbersome. So, mathematicians have come up with alternative notation schemes for successive differentiation.

Alternative Notations

For the same function, y = f(x), the following alternative notation scheme is also valid for successive differentiation (f’(x) is read is “f-prime of x”):

Apart from this, I have also seen the following notion in physics when variables are differentiated with respect to (specifically) time:

While we are on the topic of physics, why don’t we explore the applications of successive differentiation?

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Applications of Successive Differentiation

Successive differentiation primarily serves the purpose of computing what we call as higher order derivatives. While dy/dx refers to the derivative of ‘y’ with respect to ‘x’, d²y/dx² refers to the second derivative, d³y/dx³ to the third derivative, and so on.

If ‘x’ represents the location of an object as a function of time ‘t’, then dx/dt (the first derivative) represents the rate of change of location of the object with respect to time — also known as speed.

Similarly, d²x/dt² represents the rate of change of speed with respect to time — also known as acceleration. d³x/dt³ represents the rate of change of acceleration (which is the rate of change of the rate of change of the location; I apologise) — also known as jerk; I promise that DID NOT make that up.

If we keep going, we arrive at snap, crackle, and pop respectively. I have never had the need to use jerk upward though. Similarly, any variable that varies with respect to another one can have higher order variables.

In most sciences that I have had exposure to, very high order variables don’t show up. Perhaps I’m being too naive or nature’s simplicity is the cause for this.

I refer specifically to “nature” because “finance” (a human invention) is one field where I have needed relatively higher order derivatives. In fact, when people refer to a very huge number as an “astronomical” number, they are probably oblivious to the fact that astronomical numbers are nothing compared to “financial” numbers. And that is not a joke.

Final Thoughts

I think by now you would agree with me that the mathematics behind successive differentiation is not complicated. However, people often taken the notations and the reasoning behind the notations for granted.

I hope I was able to shed some light on why we use the notations we use for successive differentiation. Also, I hope that I was able to give you a rough picture about the applications of successive differentiation as well.

We are currently focusing on building strong calculus fundamentals. In the forthcoming essays in the calculus series, we will be dealing with incrementally more complex and more powerful calculus concepts. So, watch this space for more calculus essays in the future.

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Reference and credit: Silvanus Thompson.

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Further reading that might interest you:

• How To Use Science To Detect Fraud?
• The New Industrial Revolution Is Here.
• Is Zero Really Even Or Odd?

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