Defining arbitary Derivatives of functions

Question

If I have two functions like these f[a_,b_] and g[a_,b_]. Lets say

f[a_,b_]:= a^2 + b 
g[a_,b_]:= a + b

some times I encounter Dt[f[a,b] , g[a,b]] and I want to have a defined value for this derivative, like Dt[f[a,b] , g[a,b]]-> R1 How can I define new derivative for this problem in Mathematica?

I mean whenever I differentiate f with respect g I get R1.

Answer 1

protected = Unprotect[Dt]

{"Dt"}

Dt[f[___], g[___]] := R1

Dt[f[a, b], g[]]

R1

Clear your definition

Clear@Dt

Dt[f[], g[]]

0

Remember

If your definition is no longer needed Clear it - otherwise you might get unwanted results in other areas. And restore protection:

Protect[Evaluate[protected]]

{"Dt"}

Answer 2

Problem with defining such derivatives is that Dt doesn't hold its arguments, so if f and g have some definitions

ClearAll[f, g]
f[a_, b_] := a^2 + b
g[a_, b_] := a + b

then they are evaluated when passed to Dt, so "standard trick" with defining UpValues like:

f /: Dt[HoldPattern@f[a_, b_], HoldPattern@g[a_, b_]] := r1[a, b]

will not work.

What you can do is to use an "environment", in which this derivative will have desired value.

ClearAll[f, g]
f[a_, b_] := a^2 + b
g[a_, b_] := a + b

ClearAll[withMyDerivative]
SetAttributes[withMyDerivative, HoldFirst]
withMyDerivative[expr_] :=
    Block[{g},
        g /: Dt[f[a_, b_], g[a_, b_]] := r1[a, b];
        expr
    ]

Now, inside withMyDerivative environment, Dt[f[a_, b_], g[a_, b_]] will evaluate to r1[a, b].

g[a, b] (f[a, b] + Dt[f[a, b], g[a, b]]) // withMyDerivative
(* (a + b) (a^2 + b + r1[a, b]) *)