Can I ask Mathematica to replace the arguments of my functions with a replacement rule?

Question

I have an equation that I evaluate at some point (let's say $x=1$) that have terms of the form

f[1,y] D[g[1,y],{y,2}]

Is there an easy way to replace [1,y] by [x,y] with a simple replacement rule? The thing is that

g[1,y] /. g[1,y] -> g[x,y]

will indeed replace g, but it won't work for any of its derivatives, and there are far too many functions and derivatives for me to make a replacement rule for all of them.

Would there be a clever way to do this? So far, my only solution has been copy-pasting my equations in Word and using the text replacement tool in there. It would be neat not having to go outside of Mathematica to do this though.

Answer 1

Taking the question as "how to replace the first argument in $g(a,b)$ as well as all of its derivatives", you can do this:

Check the InputForm of the derivative:

D[g[1, y], {y, 2}] // InputForm

(* ==> Derivative[0, 2][g][1, y] *)

Add a corresponding pattern to the replacement rules:

f[1, y] D[g[1, y], {y, 2}] /. 
    {g[1, y] -> g[x, y], Derivative[d___][g][1, y] :> Derivative[d][g][x, y]}

Answer 2

A more general approach that replaces argument lists [1, y] anywhere (note, that this might result in unwanted replacements compared to Szabolcs's solution):

(f[1, y] D[g[1, y], {y, 2}]) /. h_[1, y] :> h[x, y]    

(* ==>  f[x, y]*Derivative[0, 2][g][x, y] *)