how to ask for user defined function using manipulate

Question

Probably this has been asked before, but I could not find an answer. I am trying to create an interactive applet using "Manipulate" that works with a user defined function. I would like to have an input field in the control part of "manipulate" and work with the expression the user enters in the input field. For example,

Manipulate[Plot[xx, {t, 0, 1}], {{xx, t^2, "x(t)="}}]

plots the graph of the expression entered in the input field, but I don't know how to modify this to graph the derivative. Any help would be appreciated.

Answer 1

It would be nice to have a "bullet-proof" Manipulate[] that does not depend on, or is not affected by, global values, such as one for t. There are a few problems, challenges and requirements:

1. The use of Global`t in the Manipulate code means users can interfere with its operation by setting t equal to something.
2. More significantly, InputField does not localize symbols. One has to do one's own localization.
3. Another requirement I sought was to be able to display the localized t as t, in both StandardForm and TraditionalForm.
4. A bigger challenge is that Manipulate rather aggressively rewrites the user's code to remap symbolic expressions in terms of localized expressions (e.g. in terms of $CellContext`t$$, which is later converted to FE`t$$nnn), while sometimes allowing global values of t to slip in. It seems bigger to me, because I have been unable to predict with 100% confidence when global t gets evaluated in the process of building the DynamicModule[] output of Manipulate[].

Well, with a bit of luck, I've got it down to the following:

t = 4;  (* to try to cause trouble *)

(* periphrasis needed, to get around Manipulate's remapping of Symbol["t"]
   Fails: PlotLabel -> ({f, df} /. HoldPattern[t] :> Symbol["t"]) *)
Manipulate[
 With[{f = ReleaseHold[
     xx /. HoldPattern[tt_ /; MatchQ[tt, Symbol["t"]]] :> t
     ]},
  With[{df = D[f, t]},                (* @lowriniak's idea *)
   (* other code as needed *)
   Plot[df, {t, 0, 1}, PlotLabel -> ({f, df})]
   ]],
 {{xx, Unevaluated[t^2], "x(t)="},    (* Manipulate does not localize initial value *)
  InputField[#, Hold[Expression]] &}, (* @Nasser's String method can be adapted, too *)
 {{t, Unevaluated[t]},                (* Manipulate does not localize initial value *)
  ControlType -> None},
 Initialization :>
  (MakeBoxes[t, form_] := FormBox["t", form])  (* Otherwise get things like FE`t$$959 *)
 ]

Answer 2

When you plot an expression, Plot substitutes in the value of the variable (in your case t) so you cannot do the derivative in the Plot (as t will no longer be a symbol, but the number substituted in). To get around this you can do the derivative outside the plot, like this:

Manipulate[
  With[
    {der = D[xx, t]}, 
    Plot[der, {t, 0, 1}]
  ], 
  {{xx, t^2, "x(t)="}}
]

Note With is used here to scope the variable der.

Answer 3

You can use InputField as string, then use ToExpression to convert to Mathematica expression, then use D to take the derivative.

Manipulate[
 tick;
 g = ToExpression[f];
 Plot[{g, Evaluate[D[g, x]]}, {x, -2 Pi, 2 Pi}, 
  PlotLegends -> {g, "its derivative"}],
 {f, Sin[x], InputField[#, String] &},
 Button["Do it", tick = Not[tick]],
 {{f, "Sin[x]"}, None},
 {{tick, False}, None},
 TrackedSymbols :> {tick}
 ]