Question about forcing derivatives

Question

I'm here with another probably simple question. So I am taking derivatives of a function with respect to another variable not used in that function. So first I define that derivative

D[r[n, k, h, L, \[Mu]], lam] = v[n, k, h, L, \[Mu]], lam]/n

So this is all well and good but then when I actually try to use that derivative in the context of another function, whoopsies it seems like mathematica has forgotten

In[773]:= D[3*r[n, k, h, L, \[Mu]], lam]

Out[773]= 0

So I checked if the original derivative is still defined and lo and behold

In[774]:= D[r[n, k, h, L, \[Mu]], lam]

Out[774]= v/n

(where v is the function of v) How do I make mathematica do this pattern matching correctly? I'm sure this has been asked before but I didn't see quite where.

I thanks to daniel have figured that using

 D[f_ : 1 r[n, k, h, L, \[Mu]], lam] ^= v/n

Gives also the wrong answer in that D[r[n, k, h, L, [Mu]], lam] gives the exact same answer as D[3*r[n, k, h, L, [Mu]], lam] which is obviously wrong

Edit: I am still unable to get this to work correctly so I've uploaded my notebook to git. You can find my full code here: https://github.com/pjdog/mathematicaStuff/blob/main/danielsonderivatives.nb

Answer 1

You can add the option NonConstants -> {r} to tell D that r does, indeed, depend on non-explicit variables. This is the best way, I think—then you can use all of D's standard rules. Otherwise, you'd need to define pattern-matching to account for every single possible way r could occur in an expression, which is...a lot. (You could probably also replace r with an "internal" function which does explicitly depend on lam, but I'll take the NonConstants route here.)

But it's a bit finicky to get it to work as intended, especially when considering that you might have other nonconstants, and other variables with respect to which you might be taking the derivative.

(I'd also recommend using upvalues to associate the definition with r instead of D. I also recommend making the arguments patterns by appending a _, otherwise only the literal symbols n, k, etc. will work!)

Here, I check if you're taking the derivative with respect to any other symbols as well; if not, I give v/m, and otherwise I apply the $\partial/\partial\lambda$ derivative first, get v/m, and then take the remaining derivatives of that.

r /: D[r[n_, k_, h_, L_, \[Mu]_], p1___Symbol, lam, p2___Symbol, NonConstants -> {x___, r, y___}] :=
       Switch[Hold[p1, p2],
                Hold[], v/m,
                _     , D[v/m, p1, p2, NonConstants -> {x, r, y}]
              ]

One could repeat this sort of procedure to account for nth-derivative syntax and vector derivatives—let me know if that would be useful!

You also might want to add r as a nonconstant to any evaluation of D. This requires modifying D.

Unprotect[D];Clear[D];

Block[{r}, 

 D[expr_, p__, NonConstants -> {x___}] :=
   D[expr, p, NonConstants -> {x, r}] /; FreeQ[{x}, r, {1}];

 D[expr_, p__, NonConstants -> x_Symbol] :=
   D[expr, p, NonConstants -> {x, r}] /; !MatchQ[x, r];

 D[expr_, p : ((_List | _Symbol)..)] :=
   D[expr, p, NonConstants -> {r}]

];

Protect[D];

Let me know if you'd like me to explain anything I use here!