Mathematica: Simplify with user-defined function

0
votes

Input:

f[x_] := Sqrt[x^2 + y^2]
f'[x]

Output:

x / Sqrt[x^2 + y^2]

How do I get Mathematica to replace the denominator by f[x] itself? (Note: this is a simple example of a more complicated differentiation problem, in which the function itself is complicated but shows up a lot in the derivative.)

That is, desired Output is:

x / f[x]

I tried

Simplify[f'[x], TransformationFunctions -> {f}]

but to no avail. Any help is appreciated!

2
wolfram-mathematica
If you do define f, it will not be possible to have it in the output as it would immediately auto-evaluate. If you type x / f[x] and press shift-enter, it'll also give you x / Sqrt[x^2 + y^2] - Szabolcs

2 Answers

1
votes

I think it's very hard to do this in general; in your specific example one can use a rule such as 

rules = {z_^2 + y^2 -> Hold[f[z]^2]};

and then

f'[x] /. rules

(* x/Sqrt[Hold[f[x]^2]] *)

f''[x] /. rules

(* -(x^2/Hold[f[x]^2]^(3/2)) + 1/Sqrt[Hold[f[x]^2]] *)

Working with the square root is more difficult and I think one rule is not enough, the basic reason being :

Sqrt[x^2 + y^2] // FullForm
1/Sqrt[x^2 + y^2] // FullForm
1
votes

I think you can do it like this:

Clear[f,g]
g[expr_]:=expr/.(x_^2+y_^2):> (f[x])^2
Simplify[D[Sqrt[x^2+y^2],x],TransformationFunctions->{Automatic,g},Assumptions->f[x]>0]

it will give x/f[x] as a result.