… The video creator actually, in all likelihood, is not thinking of hyperreals here, as he elsewhere expresses significant philosophical wariness of the "infinitely small," but I don't know if this caveat is germane to your comment.
I don't know how he introduced the first derivative or how the notation was explained, so I'm not sure whether his opinions about infinitesimals matter in his suggestion about how to think of this variable symbol: d
2f/dx
2. However, he does explain that dx is approaching zero (
ever closer), so his aversion to infinitesimals may simply be that he's more of a numerical-minded guy who doesn't want to get sucked into infinity.
You sound fairly-well versed in single-variable calculus, so I'll comment at that level.
Before the section you referenced, he explains how we get the notation for the second derivative d(df/dx)/dx (and that he thinks this end result is awkward and clumsy -- it is), and then how it becomes the usual, compact d
2f/dx
2. He then goes on to suggest a way of interpreting why the notation d
2f/dx
2 makes sense.
He takes the difference of function df's outputs at two values of x and subtracts them to find their difference d(df).
d(df
)=df
1-df
2
Then he states that this difference may be thought of as a product: some contant times the square of dx because d(df) and dx
2 are proportional.
d(df) ≈ (some constant) ∙ dx
2
As dx gets smaller and smaller, the approximation error gets smaller and smaller, until, in the limit, there is no error to speak of.
He then divides each side by dx
2
d(df)/dx
2
That process is a difference quotient; dividing a difference of outputs by a small change in input. It's just a variation on how it's usually written out. Again, he's not explaining how to calculate d
2f/dx
2. He's explaining how to think about the symbolism. And, he was thinking of the difference d(df) and dx both as differentials (very tiny numbers).
He finishes by explicity stating that d is not a variable that can be manipulated as though it's a number, but nevertheless it's just convenient and compact to write d(df) as d
2f.
That section of the video is just one way to parse the output variable for the second derivative of function f(x):
d
2f/dx
2
I read the superscript 2 on d
2f as showing a second derivative, and I pretty much ignore the dx
2, thinking of it as a marker that tells me the independent variable is x. Yet, there are times (read analysis) when it's helpful to think in terms of infinitesimals, and then for me dx
2 is the square of some tiny number.