Second Derivative Question

Metronome

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https://youtu.be/BLkz5LGWihw?t=2m29s

In the above video, is it stated that d(df) or d²f can be thought of as the difference between two consecutive output changes, each resulting from an input change of size dx. Why is a difference, rather than a quotient, the relevant expression? This is unexpected because a derivative is normally defined as a quotient, the ratio between output change and input change.
 
… a derivative is normally defined as a quotient …
"Normally", I would say that a derivative is defined as the limit of a 'difference quotient', as the change in the independent variable approaches zero. But, there are different approaches to defining derivative.

One approach treats dx not as a number, and another approach treats dx as a number. (In this latter approach, the tiny number dx is called a differential or an infinitesimal. I've also heard references to "hypernumber".) I don't know whether you're taking a math class, and I don't know what you're studying.

Are you familiar with limits? Have you seen a difference quotient?

In the video, they treat dx as a differential, and they take a difference first and divide second. They also let dx get very small (nearing zero). This seems like just a different way of explaining the "normal" method that I learned.

Without knowing more about your situation, I don't feel comfortable posting anything else. I never studied the differential approach; but I've seen it. :cool:
 
"Normally", I would say that a derivative is defined as the limit of a 'difference quotient', as the change in the independent variable approaches zero. But, there are different approaches to defining derivative.

One approach treats dx not as a number, and another approach treats dx as a number. (In this latter approach, the tiny number dx is called a differential or an infinitesimal. I've also heard references to "hypernumber".) I don't know whether you're taking a math class, and I don't know what you're studying.

Are you familiar with limits? Have you seen a difference quotient?

In the video, they treat dx as a differential, and they take a difference first and divide second. They also let dx get very small (nearing zero). This seems like just a different way of explaining the "normal" method that I learned.

Without knowing more about your situation, I don't feel comfortable posting anything else. I never studied the differential approach; but I've seen it. :cool:

I'm self-teaching at the moment, not enrolled in a course. I am close to advancing from single to multi variable calc, and am familiar with limits and difference quotients. 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.
 
… 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: d2f/dx2. 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 d2f/dx2. He then goes on to suggest a way of interpreting why the notation d2f/dx2 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)=df1-df2

Then he states that this difference may be thought of as a product: some contant times the square of dx because d(df) and dx2 are proportional.

d(df) ≈ (some constant) ∙ dx2

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 dx2

d(df)/dx2

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 d2f/dx2. 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 d2f.

That section of the video is just one way to parse the output variable for the second derivative of function f(x):

d2f/dx2

I read the superscript 2 on d2f as showing a second derivative, and I pretty much ignore the dx2, 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 dx2 is the square of some tiny number.
 
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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: d2f/dx2. 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. ;)

What I don't understand about the whole debate is, if dx is not treated as a number, but as part of an inseparable operator, symbol, etc., how would marginal analysis, substitution, or integration by parts be possible?

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 d2f/dx2. He then goes on to suggest a way of interpreting why the notation d2f/dx2 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)=df1-df2

Then he states that this difference may be thought of as a product: some contant times the square of dx because d(df) and dx2 are proportional.

d(df) ≈ (some constant) ∙ dx2

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 dx2

d(df)/dx2

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 d2f/dx2. 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 d2f.

That section of the video is just one way to parse the output variable for the second derivative of function f(x):

d2f/dx2

I read the superscript 2 on d2f as showing a second derivative, and I pretty much ignore the dx2, 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 dx2 is the square of some tiny number.

Thanks, I think I understand the intuition a bit better now.
 
What I don't understand about the whole debate is, if dx is not treated as a number, but as part of an inseparable operator, symbol, etc., how would marginal analysis, substitution, or i͏ntegration by parts be possible?
Those topics generally come later. When calculus is first introduced, dx is not treated as a number. Once students get used to concepts and notations (like dx and d/dx), the idea of differentials can be introduced; symbols dx and dy are then manipulated as infinitely-small numbers.

It's two different approaches to notation, and we use each depending on context.

I still remember my Calculus I (derivatives) instructor emphasizing throughout that dx is NOT a number. On the first day of Calculus II (integrals), the new instructor told us (d/dx)(y) is changing to dy/dx, and that this is a ratio of infinitely-small numbers. It was confusing! Later, I had wished the introductory course would have demonstrated both notations at the very beginning and warned students to expect and distinguish between these viewpoints.

As students, we all need to be flexible (even when the instruction is not). :cool:
 
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