Replace one order of derivative inside higher one

How could I replace for example:

Derivative[f][x]

by

Derivative[g][x]

where

g[x]=Derivative[f][x]

?

But doing this in a manner that work for all cases (all combinations of mixed derivates and multiple variables)?

This seems to be very simple:

With[{f = Derivative[-1][g]},
Derivative[f][x]]

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

The objective of this was to use replacement rules with lower derivatives results inside higher ones.

Transform[differentialEquationArg_]:=Module[{differentialEquation,previousDerivative},
differentialEquation=differentialEquationArg//.Derivative[a__][b_][c__]/;(previousDerivative=GetPreviousDerivative[{a},Length[{a}]];Total[{a}]>1&&True):>Derivative[previousDerivative/.List->Sequence][Derivative[{a}-previousDerivative/.List->Sequence][b][c]][c];
differentialEquation
]

GetPreviousDerivative[derivativeTupleArg_,workingPositionArg_]:=Module[{derivativeTuple,workingPosition},
derivativeTuple=derivativeTupleArg;
workingPosition=workingPositionArg;
If[derivativeTuple[[workingPosition]]>0,derivativeTuple[[workingPosition]]-=1,workingPosition-=1;derivativeTuple=GetPreviousDerivative[derivativeTuple,workingPosition]];
derivativeTuple
]

Example:

D[y[t],t,t]+y[t]+D[y[t],t]==0

Out: y''(t)+y'(t)+y(t)=0

Transform[%]

Out: y'(t)'(t)+y'(t)+y(t)=0

%//FullForm

Equal[Plus[y[t],Derivative[y][t],Derivative[Derivative[y][t]][t]],0]

Not completely tested, can have some bugs.