Get a "step-by-step" evaluation in Mathematica

Question

Is it possible in Mathematica to get a step-by-step evaluation of some functions; that's to say, outputting not only the result but all the stages that have led to it? If so, how does one do it?

Example : Let's say I want to know the steps to get the derivative of $\cos x\times\exp x$; it should first tell me that it's equal to $\frac{d}{dx}(\exp x)\times\cos x+\exp x \times \frac{d}{dx}(\cos x)$ and then render the result to say $\exp{x}\times(\cos x-\sin x)$.

Answer 1

For differentiation at least, old versions of Mathematica had a demonstration function called WalkD[] that holds your hand and shows what is done at each stage up until the final answer.

In general, however...

You should realize at the outset that while knowing about the internals of Mathematica may be of intellectual interest, it is usually much less important in practice than you might at first suppose.

Indeed, one of the main points of Mathematica is that it provides an environment where you can perform mathematical and other operations without having to think in detail about how these operations are actually carried out inside your computer.

...

Particularly in more advanced applications of Mathematica, it may sometimes seem worthwhile to try to analyze internal algorithms in order to predict which way of doing a given computation will be the most efficient. And there are indeed occasionally major improvements that you will be able to make in specific computations as a result of such analyses.

But most often the analyses will not be worthwhile. For the internals of Mathematica are quite complicated, and even given a basic description of the algorithm used for a particular purpose, it is usually extremely difficult to reach a reliable conclusion about how the detailed implementation of this algorithm will actually behave in particular circumstances.

A typical problem is that Mathematica has many internal optimizations, and the efficiency of a computation can be greatly affected by whether the details of the computation do or do not allow a given internal optimization to be used.

Put another way: how Mathematica does things doesn't necessarily correspond to "manual" methods.

---

Here's my modest attempt to (somewhat) modernize WalkD[]:

Format[d[f_, x_], TraditionalForm] := DisplayForm[RowBox[{FractionBox["\[DifferentialD]",
                                                  RowBox[{"\[DifferentialD]", x}]], f}]];

SpecificRules = {d[(f_)[u___, x_, v___], x_] /;
                 FreeQ[{u}, x] && FreeQ[{v}, x] :> D[f[u, x, v], x],
                 d[(a_)^(x_), x_] :> D[a^x, x] /; FreeQ[a, x]};

ConstantRule = d[c_, x_] :> 0 /; FreeQ[c, x];

LinearityRule = {d[f_ + g_, x_] :> d[f, x] + d[g, x],
                 d[c_ f_, x_] :> c d[f, x] /; FreeQ[c, x]};

PowerRule = {d[x_, x_] :> 1, d[(x_)^(a_), x_] :> a*x^(a - 1) /; FreeQ[a, x]};

ProductRule = d[f_ g_, x_] :> d[f, x] g + f d[g, x];

QuotientRule = d[(f_)/(g_), x_] :> (d[f, x]*g - f*d[g, x])/g^2;

InverseFunctionRule = d[InverseFunction[f_][x_], x_] :>
                      1/f'[InverseFunction[f][x]];

ChainRule = {d[(f_)^(a_), x_] :> a*f^(a - 1)*d[f, x] /; FreeQ[a, x],
             d[(a_)^(f_), x_] :> Log[a]*a^f*d[f, x] /; FreeQ[a, x],
             d[(f_)[g__], x_] /; ! FreeQ[{g}, x] :>
             (Derivative[##][f][g] & @@@ IdentityMatrix[Length[{g}]]).(d[#, x] & /@ {g}),
             d[(f_)^(g_), x_] :> f^g*d[g*Log[f], x]};

$RuleNames = {"Specific Rules", "Constant Rule", "Linearity Rule", "Power Rule",
              "Product Rule", "Quotient Rule", "Inverse Function Rule", "Chain Rule"};

displayStart[expr_] := CellPrint[
  Cell[BoxData[MakeBoxes[HoldForm[expr], TraditionalForm]], "Output", 
   Evaluatable -> False, CellMargins -> {{Inherited, Inherited}, {10, 10}}, 
   CellFrame -> False, CellEditDuplicate -> False]]

displayDerivative[expr_, k_Integer] := CellPrint[
  Cell[BoxData[TooltipBox[RowBox[{InterpretationBox["=", Sequence[]], "  ", 
       MakeBoxes[HoldForm[expr], TraditionalForm]}], $RuleNames[[k]], 
     LabelStyle -> "TextStyling"]], "Output", Evaluatable -> False, 
   CellMargins -> {{Inherited, Inherited}, {10, 10}}, 
   CellFrame -> False, CellEditDuplicate -> False]]

WalkD[f_, x_] := Module[{derivative, oldderivative, k}, 
        derivative = d[f, x]; displayStart[derivative];
        While[! FreeQ[derivative, d],
            oldderivative = derivative; k = 0;
            While[oldderivative == derivative,
                      k++;
                      derivative = derivative /. 
                              ToExpression[StringReplace[$RuleNames[[k]], " " -> ""]]];
            displayDerivative[derivative, k]];
        D[f, x]]

I've tried to make the formatting of the derivative look a bit more traditional, as well as having the differentiation rule used be a tooltip instead of an explicitly generated cell (thus combining the best features of WalkD[] and RunD[]); you'll only see the name of the differentiation rule used if you mouseover the corresponding expression.

Answer 2

Use:

http://www.wolframalpha.com/input/?i=D[Cos[x]*Exp[x]%2C+x]

and select "Show steps". More in Mathematica would be

WolframAlpha["D[Cos[x]*Exp[x], x]"]

or even

WolframAlpha["D[Cos[x]*Exp[x], x]", IncludePods -> "Input", 
 AppearanceElements -> {"Pods"}, PodStates -> {"Input__Show steps"}]

Answer 3

To see the steps for taking indefinite integrals one can use free rule-based integrator nicknamed Rubi crafted by Albert D. Rich:

Click on the sample integration problem at the end of the notebook and press Shift-Enter to evaluate it. After a minute or so depending on the speed of your computer, the first step of the integration should be displayed. To see successive steps, click on the intermediate results and press Shift-Enter.

In many cases this integrator produces terser output than the built-in.

Answer 4

here is a function based on WolframAlpha[]

ShowSteps[exp_] := 
  WolframAlpha[ ToString@HoldForm@InputForm@exp, 
  {{"Input", 2}, "Content"},  PodStates -> {"Input__Show steps"}]

SetAttributes[ShowSteps, HoldAllComplete]

for limits use

PodStates -> {"Limit__Show steps"}

for integration

PodStates -> {"IndefiniteIntegral__Show steps"} 

Update:

WolframAlpha changed output.

Now ShowSteps should work with:

ShowSteps[exp_] := 
  WolframAlpha[ToString@HoldForm@InputForm@exp, 
  {{"Input", 1}, "Content"}, 
   PodStates -> {"Input__Step-by-step solution","Input__Show all steps"}]

 SetAttributes[ShowSteps, HoldAll]

Update - Nov 2014

ShowSteps[exp_] := 
 WolframAlpha[ToString@HoldForm@InputForm@exp, 
 {{"Input", 2}, "Content"}, 
 PodStates -> {"Input__Step-by-step solution"}]

SetAttributes[ShowSteps, HoldAll]

Answer 5

I have improved J. M.'s version of walkD by adding error handling. I have also added walkInt that works like walkD except for integration. Code:

Format[d[f_, x_], TraditionalForm] := Module[{paren, boxes},
    paren = MatchQ[f,Plus[_,__]];
    boxes = RowBox[{f}];
    If[paren,
        boxes = RowBox[{"(", boxes, ")"}]
    ];
    boxes = RowBox[{FractionBox["\[DifferentialD]", RowBox[{"\[DifferentialD]", x}]], boxes}];
    DisplayForm[boxes]
];

dSpecificRules = {d[x_, x_] :> 1, d[(f_)[x_], x_] :> D[f[x], x],
                 d[(a_)^(x_), x_] :> D[a^x, x] /; FreeQ[a, x]};

dConstantRule = d[c_, x_] :> 0 /; FreeQ[c, x];

dLinearityRule = {d[f_ + g_, x_] :> d[f, x] + d[g, x],
                 d[c_ f_, x_] :> c d[f, x] /; FreeQ[c, x]};

dPowerRule = {d[x_, x_] :> 1, d[(x_)^(a_), x_] :> a*x^(a - 1) /; FreeQ[a, x]};

dProductRule = d[f_ g_, x_] :> d[f, x] g + f d[g, x];

dQuotientRule = d[(f_)/(g_), x_] :> (d[f, x]*g - f*d[g, x])/g^2;

dInverseFunctionRule := d[InverseFunction[f_][x_], x_] :>
                      1/Derivative[1][f][InverseFunction[f][x]];

dChainRule = {d[(f_)^(a_), x_] :> a*f^(a - 1)*d[f, x] /; FreeQ[a, x],
             d[(a_)^(f_), x_] :> Log[a]*a^f*d[f, x] /; FreeQ[a, x],
             d[(f_)[g_], x_] :> (D[f[x], x] /. x -> g)*d[g, x],
             d[(f_)^(g_), x_] :> f^g*d[g*Log[f], x]};

$dRuleNames = {"Specific Rules", "Constant Rule", "Linearity Rule", "Power Rule",
              "Quotient Rule", "Product Rule", "Inverse Function Rule", "Chain Rule"};

displayStart[expr_] := CellPrint[
    Cell[BoxData[MakeBoxes[HoldForm[expr], TraditionalForm]], "Output", 
        Evaluatable -> False, CellMargins -> {{Inherited, Inherited}, {10, 10}}, 
        CellFrame -> False, CellEditDuplicate -> False]];

displayDerivative[expr_, k_Integer] := CellPrint[
    Cell[BoxData[TooltipBox[RowBox[{InterpretationBox["=", Sequence[]], "  ", 
        MakeBoxes[HoldForm[expr], TraditionalForm]}], "Differentation: " <> $dRuleNames[[k]], 
        LabelStyle -> "TextStyling"]], "Output", Evaluatable -> False, 
    CellMargins -> {{Inherited, Inherited}, {10, 10}}, 
    CellFrame -> False, CellEditDuplicate -> False]];

walkD::differentationError = "Failed to differentiate expression!";

walkD[f_, x_] := Module[{derivative, oldderivative, k}, 
    derivative = d[f, x]; displayStart[derivative];
    While[! FreeQ[derivative, d],
        oldderivative = derivative; k = 0;
        While[oldderivative == derivative,
            k++;
            If[k > Length@$dRuleNames,
				Message[walkD::differentationError];
				Return[D[f, x]];
			];
			derivative = derivative /. ToExpression["d" <> StringReplace[$dRuleNames[[k]], " " -> ""]]
        ];
        displayDerivative[derivative, k]];
    D[f, x]
];


Format[int[f_,x_],TraditionalForm]:= (
    paren = MatchQ[f,Plus[_,__]];
    boxes = RowBox[{f}];
    If[paren,
        boxes = RowBox[{"(", boxes, ")"}]
    ];
    boxes = RowBox[{boxes, "\[DifferentialD]", x}];
    boxes = RowBox[{"\[Integral]", boxes}];
    DisplayForm[boxes]
);

intSpecificRules = {int[(f_)[x_], x_] :> Integrate[f[x], x],
                   int[(a_)^(x_), x_] :> Integrate[a^x, x] /; FreeQ[a, x]};

intConstantRule = int[c_, x_] :> c*x /; FreeQ[c, x];

intLinearityRule = {int[f_ + g_, x_] :> int[f, x] + int[g, x],
                   int[c_ f_, x_] :> c int[f, x] /; FreeQ[c, x]};

intPowerRule = {int[x_, x_] :> x^2 / 2, int[1/x_, x_] :> Log[x], int[(x_)^(a_), x_] :> x^(a + 1)/(a + 1) /; FreeQ[a, x]};

intSubstitutionRule = {
                        int[(f_)^(a_), x_] :> ((Integrate[u^a, u] / d[f, x]) /. u -> f) /; FreeQ[a, x] && FreeQ[D[f, x], x],
                        int[(f_)^(a_) g_, x_] :> ((Integrate[u^a, u] / d[f, x]) * g /. u -> f) /; FreeQ[a, x] && FreeQ[FullSimplify[D[f, x] / g], x],
                        int[(a_)^(f_), x_] :> (a ^ f)/(d[f, x] * Log[a]) /; FreeQ[a, x] && FreeQ[D[f, x], x],
                        int[(a_)^(f_) g_, x_] :> (a ^ f)/(d[f, x] * Log[a]) * g /; FreeQ[a, x] && FreeQ[FullSimplify[D[f, x] / g], x],
                        int[(f_)[g_], x_] :> (Integrate[f[u], u] /. u -> g) / d[g, x] /; FreeQ[D[g, x], x],
                        int[(f_)[g_] h_, x_] :> (Integrate[f[u], u] /. u -> g) / d[g, x] * h /; FreeQ[FullSimplify[D[g, x] / h], x]
                    };

intProductRule = int[f_ g_, x_] :> int[f, x] g - int[int[f, x] * d[g, x], x];

$intRuleNames = {"Specific Rules", "Constant Rule", "Linearity Rule", "Power Rule", "Substitution Rule", "Product Rule"};

displayIntegral[expr_, k_Integer] := CellPrint[
  Cell[BoxData[TooltipBox[RowBox[{InterpretationBox["=", Sequence[]], "  ", 
       MakeBoxes[HoldForm[expr], TraditionalForm]}], "Integration: " <> $intRuleNames[[k]], 
     LabelStyle -> "TextStyling"]], "Output", Evaluatable -> False, 
   CellMargins -> {{Inherited, Inherited}, {10, 10}}, 
   CellFrame -> False, CellEditDuplicate -> False]];

walkInt::integrationError = "Failed to integrate expression!";
walkInt::differentationError = "Failed to differentiate expression!";

walkInt[f_, x_] := Module[{integral, oldintegral, k, leafcounts, ruleused},
    integral = int[f, x]; displayStart[integral];
    leafcounts = {};
    ruleused = "";
    While[! FreeQ[integral, int],
        If[ruleused == "Product Rule",
            AppendTo[leafcounts, LeafCount @ integral];
            If[Length @ leafcounts >= 5 && OrderedQ @ Take[leafcounts, -5],
                Message[walkInt::integrationError];
                Return[Integrate[f, x]];
            ];
        ];
        oldintegral = integral; k = 0;
        While[oldintegral == integral,
            k++;
            If[k > Length@$intRuleNames,
				Message[walkInt::integrationError];
				Return[Integrate[f, x]];
			];
			integral = integral /. ToExpression["int" <> StringReplace[$intRuleNames[[k]], " " -> ""]]
        ];
        ruleused = $intRuleNames[[k]];
		displayIntegral[integral, k];
		While[! FreeQ[integral, d],
			oldintegral = integral; k = 0;
			While[oldintegral == integral,
				k++;
				If[k > Length@$dRuleNames,
                    Message[walkInt::differentationError];
                    Return[Integrate[f, x]];
                ];
                integral = integral /. ToExpression["d" <> StringReplace[$dRuleNames[[k]], " " -> ""]]
            ];
            displayDerivative[integral, k]];
        ];
    Integrate[f, x]
];

Sample output:

Answer 6

I found this answer from Quora by Vitaliy Kaurov: https://www.quora.com/Can-Mathematica-show-me-step-by-step-solutions

Wolfram|Alpha (W|A) is integrated in Wolfram Language (WL or Mathematica) via function WolframAlpha that can place API calls to W|A servers. So you can pull step-by-step solutions into your notebook via simple WL code like:

WolframAlpha[
  "integrate 1/(1+cos(x)^2)", 
  {{"IndefiniteIntegral", 2}, "Content"}, 
  PodStates -> {"IndefiniteIntegral__Step-by-step solution"}]

This works well for me.

Answer 7

The output of

WolframAlpha["D[Cos[x]*Exp[x], x]", IncludePods -> "Input", 
 AppearanceElements -> {"Pods"}, PodStates -> {"Input__Show steps"}]

is