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complex - Differentiation of an unknown function


I have to take the partial differentiation of an unknown function. For example, take the unknown function to be $g(x)$. Then it's derivative w.r.t $x$ is $g'(x)$.


By default, Mathematica differentiates the function. I want to keep the result of differentiation as $d(g(x))$ and not $g'(x)$. Is there any way to achieve this?


More precisely, I am using Conjugate[g[x]] as the unknown function and I want the output should be displayed only as d[Conjugate[g[x]] and not as Conjugate'[x]g'[x].


Also, can I handle the conjugate more efficiently than just carrying it all along in the code?




Answer



Edited because the goal was changed in the comment:


This can be done by directly defining the outcome of Derivative when applied to g in the two combinations that you seem to be interested in:


Derivative[1][g][x_] := d[g[x]]

Derivative[1][Conjugate][g[x_]] := Conjugate[d[g[x]]]/d[g[x]];
Derivative[1][Conjugate][d[x_]] := Conjugate[d[d[x]]]/d[d[x]]

Derivative[1][d][x_] := d[d[x]]/d[x];
Derivative[1][d][x_Symbol] := d[d[x]]


On the second line, I used the fact that g is a generic function whose derivative under a Conjugate by default invokes the chain rule. All I do then is to reverse the chain rule by dividing by the factor d[g[x]] that the chain rule will produce. This leaves only the factor I want, and I then replace that by the desired outcome d[Conjugate[g[x]]].


The analogous thing is done for d to allow higher derivatives. The exception is when d[x] is encountered where x is the differentiation variable (which isn't in the question, but I expect may happen). Then there is no chain rule needed, and I therefore specify a separate rule for it with the pattern x_Symbol.


Here is the test:


D[g[x], x]

(* ==> d[g[x]] *)

D[Conjugate[g[x]], x]


(* ==> d[Conjugate[g[x]]] *)

D[g[x], x, x]

(* ==> d[d[g[x]]] *)

D[d[g[x]], x]

(* ==> d[d[g[x]]] *)


D[d[x], x]

(* ==> d[d[x]] *)

D[Conjugate[g[x]], x]

(* ==> Conjugate[d[g[x]]] *)

D[Conjugate[g[x]], x, x]


(* ==> Conjugate[d[d[g[x]]]] *)

Now the remaining issue is to replace the repeated application of d by formatting of the type d^2 g[x] for d[d[g[x]]]. I'll wait to see if this is really desired before doing it.


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