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RE: How do I see justification of solution?

  • To: mathgroup at
  • Subject: [mg28390] RE: [mg28377] How do I see justification of solution?
  • From: "David Park" <djmp at>
  • Date: Sun, 15 Apr 2001 00:13:41 -0400 (EDT)
  • Sender: owner-wri-mathgroup at


You can't always get Mathematica to show all the steps it used in a command
because many of the steps are buried in C code.

Still, many students and teachers would like to do and show steps which
Mathematica does automatically. One solution to this problem is a package
called Algebra`ExpressionManipulation which is available at my web site
below. The package was developed by Ted Ersek and myself to handle just this
type of problem. (I believe that Ted also has some overlapping packages
available at MathSource.) This is how we would apply it to your example to
show all the steps, complete with annotation.


HoldForm[D[Sin[x^2 + 5*x], x]]
Print["Use of chain rule."]
%% /. HoldPattern[D[(f_)[e_], x]] :>
   Derivative[1][f][e]*D[e, x]
Print["Function derivative."]
Print["Use of sum rule."]
%% /. HoldPattern[D[a_ + b_, x]] :> D[a, x] + D[b, x]
Print["Use of constant rule"]
%% /. HoldPattern[D[(a_)?(FreeQ[#1, x] & )*b_, x]] :>
   a*D[b, x]
Print["Evaluating derivatives."]

The output, which looks much better in standard form (for instance the
HoldForm is invisible), is:

HoldForm[D[Sin[x^2 + 5*x], x]]
Use of chain rule.
HoldForm[Derivative[1][Sin][x^2 + 5*x]*D[x^2 + 5*x, x]]
Function derivative.
HoldForm[Cos[5*x + x^2]*D[x^2 + 5*x, x]]
Use of sum rule.
HoldForm[Cos[5*x + x^2]*(D[x^2, x] + D[5*x, x])]
Use of constant rule
HoldForm[Cos[5*x + x^2]*(D[x^2, x] + 5*D[x, x])]
Evaluating derivatives.
HoldForm[Cos[5*x + x^2]*(2*x + 5*1)]
(5 + 2*x)*Cos[5*x + x^2]

The ExpressionManipulation package has routines for the controlled
evaluation of expressions. It also has a PositionsPalette which allows one
to highlight a subexpression and obtain its position. There is another
routine called ColorPositions which will color and label explicit positions
or positions which match a pattern. It also has a feature of "extended
positions". An extended position is a subset of level parts in an
expression. An example would be a + c in f[1+a+b+c+d]. Extended positions
can also be manipulated.

There is also an EvaluationTutorial that illustrate the routines and show
techniques for controlled expression manipulation. There are examples from
simple fraction problems to integration by parts. I like your example so
much, I might just add it to the tutorial.

David Park
djmp at

> -----Original Message-----
> I want to see how Mathematica solves the problem I give it;  Not
> just the sub-expressions that Trace puts out.
> Ex.  If I give it Find the derivative of D[sin(x^2 + 5x),x]
> I'd like to see:
>    Chain Rule (...)
>    sum Rule (x^2 + 5x)
>    power rule (x^2)
>    .
> ..
> ..
> Isn't there some way to get the thing to show the steps it took to solve
> a problem?

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