Your question is sparse in important details.

Where does Sx come from?

If Sx comes from dsolve(...,numeric) then you might be able to use that command's events facilities. (For example, you might make it halt when the derivative falls under a target value, has a discontinuous drop, etc.)

Why not provide the details?

@Carl Love No problem, I wasn't complaining. I was on my own phone and couldn't remember or check.

ps. There's another form for elementwise -- a prefix form I suppose it could be called, with an indexed name. Eg, for the OP's list of Vectors, fasteners,

    `~`[`~`[convert]](fasteners, unit_free)

Now I'm wondering how that nested `~`[`~`[convert]] might be programmatically folded together for a specified depth (here, two) -- even if in an ugly manner using `?[]` .

@Carl Love Did I forget to mention that one? (I thought that I had...)

Please upload and attach either a file with the data or a worksheet that can generate it.

@Pi Where else but the name c[i] do w or t_pe depend upon i?

What aspect do you think would change, by varying the value of i?

@Pi What expression(s) do you want to differentiate with respect to, and with respect to what?

I find those fundamental aspects unclear from your phrasing.

What is "soltion(1)"?

By "appro_function(1)" do you mean appro_function[1] ?

Please stop posting duplicates of this problem.

If you have difficutlies or issue with the answers given, or additional details, then please add them to this Question thread (using the Reply button) instead of starting wholy separate new Question threads.

@C_R No, please don't spawn a separate Question thread for such a close variant of the same topic you started.

I don't think that the special name form, eg. `#mrow(...);`  gets mspace respected when rendering. Hence the answer to your followup query might be no, not a name.

That's why above I showed a variant with a print-extension for an unevaluated function call.

@C_R Your requirements have changed.

Here is one very crude way to "nudge the Y up a bit". It's not very flexible.

The results render better in my actual Maple GUI (2023.0), without being cut off on the left.

This is all pretty hackish, because it wasn't designed for arbitrary layout. So, YMMV.

Download ts_negmspace2.mw

You should explain how you want the surface data that is to be exported, ie. your desired format.

For example, do you want it in three columns (x-y-z triples)?

Or could it just be the z-values (heights), since the x&y values follow the pattern of regularly sampling from their corresponding ranges.

You specified certain step-sizes in the call to pdsolve. But you also asked about exporting the data from the 3d plot (which by default has a different resolution). You should be clear about which spacing of x- and t-values you want for the exported data.

Or perhaps you have some other format in mind. You should be clear about what you want here.

Please don't spawn a separate Question thread for supplying the details.

@C_R Please don't spawn a wholly separate new Question thread on the issue of what your sheet looks like in M2015.

@Michael If you know that you only need to deal with a single factor then you could avoid the unnecessary expense of a complete factorization (eg. using ifactors) and just do divisions.

I would have suggested this earlier but the OP didn't originally give such details of the goal.

Please put close followups on your underlying problem in this (or one of your related previous Question threads), instead of spawning whooly new and separate Question threads for your problem.

@nicolesharp100 Please do not post a duplicate of this in an alternate Mapleprimes forum.

Feel free to submit an SCR (software change request).

@Linhuchong As Tom mentioned, there are several ways. I like his use of geometry.

I also like vv's approach.

For fun, keeping with implicitplot for no particular reason (and now more clumsily calling solve twice),

restart;
f:=2.96736996560705*10^(-12)*p^2+1.31319840299485*10^(-13)*t^2
   -8.89549693662593*10^(-7)*p+8.53128393394231*10^(-7)*t
   -3.65558815509970*10^(-30)*p*t-1:
P:=remove(has,solve(f<0,[p,t])[],t);
T:=remove(has,solve(f<0,[t,p])[],p);
p1,p2:=rhs(P[1]),lhs(P[2]);
t1,t2:=rhs(T[1]),lhs(T[2]);
plots:-implicitplot(f,p=p1..p2,t=t1..t2,scaling=constrained);