Can a variable be dependent and independent at the same time ? (On The Chain Rule)

[Moamen]

According to Keisler's Elementary Calculus: an infinitesimal approach
Chapter 2 / Section 2.6 / Pages 88-92
( https://www.math.wisc.edu/~keisler/calc.html )

We have two cases for the Chain Rule, which are: (page 89)

Case 1: dy / dt = dy / dx * dx / dt

where X is independent in dy/dx and t is independent both in dy/dt and dx/dt.

It SEEMS to me that X here, is independent to y, and dependent on t.

Can a variable be dependent and independent at the same time ?

Case 2: dy / dx = (dy/dt) / (dx/dt)

where t is the only independent variable.

If that is the case, then what is the relation between x and y called ? In other words, how can y be a function of x here when both x and y are dependent ?

And finally, if y = f(x) has an inverse function x = g(y), is x independent in both the original function and its inverse ?

 

[pka]

According to Keisler's Elementary Calculus: an infinitesimal approach
Chapter 2 / Section 2.6 / Pages 88-92
( https://www.math.wisc.edu/~keisler/calc.html )

We have two cases for the Chain Rule, which are: (page 89)

Case 1: dy / dt = dy / dx * dx / dt

where X is independent in dy/dx and t is independent both in dy/dt and dx/dt.

It SEEMS to me that X here, is independent to y, and dependent on t.

Can a variable be dependent and independent at the same time ?

Case 2: dy / dx = (dy/dt) / (dx/dt)

where t is the only independent variable.

If that is the case, then what is the relation between x and y called ? In other words, how can y be a function of x here when both x and y are dependent ?

And finally, if y = f(x) has an inverse function x = g(y), is x independent in both the original function and its inverse ?

I have an actual hardbound copy of that text in front of me as I type. I think that you may be conflating discussion material on page 88 beginning with \(\displaystyle \dfrac{dy}{dt}=\dfrac{dy}{dx}\dfrac{dx}{dt}\) with a discussion about \(\displaystyle dy=\dfrac{dy}{dx}dx\) down that page and with more discussion on page 89.

Tell us the exact example where this confusion gets in you way. Otherwise, just carry on and see if it works itself out. If not, please post again with a detailed reference.

 

[Moamen]

Thank you for your reply.

I will try to be as precise as possible. Thefollowing page numbers belong to the Free Online Edition, I don't know if it'sthe same as your edition.

(Page 85, Section 2.6 Chain Rule)
When I first read the paragraphs under thistitle,

I assumed that these paragraphs were just sayingthat, whenever we have

x = f(t), y = G(x) and y = g(t)

this means that t is the only independentvariable such that when t varies, x varies making y vary as well, and x is justa "link" ( I don't know how to put it in proper terms) between y andt.

Then (On page 87, below Example 2, Chain RuleWith Dependent Variables)

I came across that lines ( below (i) and (ii) )that say,

"where dx/dt, dy/dt are computed with t asthe independent variable, and dy/dx is computed with x as the independentvariable"

This is the first time I've seen something likethis, which is why it doesn't make any sense to me. I don't know how thisworks. I mean, when t changes, f(t) or x changes, But then how can f(t) or xchange independently ? How can a variable be both dependent and independent ATthe same time ?
Now, (On page 88, below Example 2 Continued (ii))

The author discusses the issue again, by sayingthat

dy / dt = dy/dx . dx/dt is trivial when t is theindependent variable, I guess this refers to my initial understanding of thenotation, that x is just a link between y and t. He also says that "thetwo dx's have different meanings", so my assumption is that

dx / dt : dx here is the output change due tothe infinitesimal input change, dt.
dy / dx: dx here is the infinitesimal inputchange that is not related to the dx above.

If this is incorrect, then what are thedifferent meanings?

(Page 89, Below Power Rule)

The second type of applications of the ChainRule is when, both x and y depend on a third variable t. So I'm guessing thedifference here is that y here directly depends on t, and is not related to tthrough x. But does that mean that t is the only independent variable in thiscase? I mean, how is y related to x, is x still independent to y ?

I still have a lot more questions, but this post is getting too lengthy.

 

[Moamen]

I also sent you a private message a couple of weeks ago, concerning other questions about hyperreal numbers , could you please take a look.