In passing I came across this expression the other day,

To put it in words, we have a composite function: the sin of the sin of the sin(x).

I like this expression quite a lot and for a few reasons. One reason has to do with its cascading quality, especially after we differentiate it. (We’ll look at this in a moment). As I’ll comment in just a moment, to the eye it may seem much more complicated than it is. But really, we could nest more sin’s into the expression and it doesn’t really change much when we go to differentiate it.

Another reason I like this expression is because I think it serves a nice lesson. When one first looks at it, with a mind that in our case it requires differentiation, the expression might first seem daunting. It may even evoke a sense of fear. But its derivative is actually very simple, so simple that it sort of reminds me of a coffin problem (a seemingly difficult problem with a relatively simple solution).

If I were to teach a calculus course at university at some point in the future, I would present my students with this expression on the first day for this very reason. The intention would not be a sadistic one, but to show that often in mathematics we are presented with difficult looking expressions or equations or problems – the moral being that we ought not to fear. I often find that in mathematics, and certainly also frontier mathematical physics, one can’t be ruled by fear of the daunting or of the unknown. You have to venture forward. The key is to take a deep breath and think it through step by step, experiment and just freely explore the maths.

With that said, let’s now take the derivative of the above expression. The important thing is to first identify that we need to use the chain rule, then work from left to right step by step.

Notice when we take the derivative, the outermost sin becomes cos of everything inside the parenthesis. The middle sin becomes cos of the innermost sin(x). And, finally, the innermost sin(x) differentiates to cos(x).