How to modify the code to make the function image clear?

Questions
,
1

How to modify the code so that the abscissa and ordinate of the original function and derivative function are adaptive in the same coordinate system so that the displayed image size is appropriate and the form of the function image can be felt as a whole

ClearAll@f;

f[x_] := Log[x]/-x^2

Manipulate[Refresh[functions = Table[D[f[x], {x, n}], {n, 0, nMax, 1}];
  orders = 
   Table[D[f[x], {x, n}] // Inactivate // TraditionalForm // 
     ToString, {n, 0, nMax, 1}];
  labels = 
   MapThread[#1 <> " = " <> ToString[#2, TraditionalForm] &, {orders, 
     functions}];];
 Plot[functions, {x, 0, 1}, PlotLabels -> labels, PlotRange -> All, 
  ImageSize -> 700, AspectRatio -> 1, 
  PlotLabel -> Row[{"f(x) = ", f[x]}]], {{nMax, 1, "Order"}, 1, 10, 1,
   PopupMenu}]
2

The previous effect is like this, which obviously does not meet the requirements

20221117154053.png
20221117154053.png1814×806 24.8 KB

3

The problem is that the functions depend inversely on x so they diverge as x -> 0. Try removing PlotRange -> All.

4

ClearAll@f;

f[x_] := Log[x]/-x^2
Manipulate[Refresh[functions = Table[D[f[x], {x, n}], {n, 0, nMax, 1}];
  orders = 
   Table[D[f[x], {x, n}] // Inactivate // TraditionalForm // 
     ToString, {n, 0, nMax, 1}];
  labels = 
   MapThread[#1 <> " = " <> ToString[#2, TraditionalForm] &, {orders, 
     functions}];];
 Plot[functions, {x, 0, 13}, PlotLabels -> labels, ImageSize -> 700, 
  AspectRatio -> 1, 
  PlotLabel -> Row[{"f(x) = ", f[x]}]], {{nMax, 1, "Order"}, 1, 10, 1,
   PopupMenu}]
5

20221118160821.png
20221118160821.png1322×880 35.6 KB

6

Set to equal horizontal and vertical coordinates

ClearAll@f;

f[x_] := x Log[x]
Manipulate[Refresh[functions = Table[D[f[x], {x, n}], {n, 0, nMax, 1}];
  orders = 
   Table[D[f[x], {x, n}] // Inactivate // TraditionalForm // 
     ToString, {n, 0, nMax, 1}];
  labels = 
   MapThread[#1 <> " = " <> ToString[#2, TraditionalForm] &, {orders, 
     functions}];];
 Plot[functions, {x, 0, 5}, PlotLabels -> labels, ImageSize -> 700, 
  AspectRatio -> Automatic, 
  PlotLabel -> Row[{"f(x) = ", f[x]}]], {{nMax, 1, "Order"}, 1, 10, 1,
   PopupMenu}]