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I am taking the derivative of a function with respect to another function, e.g.

D[g[f[x,y]],f[x,y]]

This outputs g'(f(x,y)), as expected. How can I specify a rule for this derivative?

I've tried the following:

Derivative[g[f[x_,y_]][f[x_, k_]]] := f[y, k]

but apparently this isn't the output of D[].

Any ideas?

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  • $\begingroup$ Try: Derivative[g[f[x_, y_]][f[x_, k_]]] := g'[f[y, k]] $\endgroup$ Jan 29, 2023 at 13:20
  • $\begingroup$ This doesn't appear to do anything.... $\endgroup$
    – Akoben
    Jan 29, 2023 at 14:04

1 Answer 1

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Assuming that you have a random expression

expr = D[g[f[x, y]], f[x, y]] + 2 x + 2 f[x, y] + g[f[x, y]] + 
   Log[D[g[f[x, y]], f[x, y]]];

that looks like this

expr

exp

you can write a replacement rule like so:

rule = Derivative[1][g][f[x, y]] -> 
    HHH[x, y]; 

and then

expr /. rule

gives

res

To understand the above, I suggest that you take a look at the FullForm of the expression for which you wanted to set a rule originally, i.e D[g[f[x, y]], f[x, y]] // FullForm

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