Replace expressions with their names in Mathematica

Question

I have some expressions in Mathematica that are defined in terms of other expressions. I want to take some functions of the larger expression and then get the result in terms of the subexpressions. Example:

In[78]:= e1 = x + y;
e2 = 2^e1;

In[80]:= D[e2, x]

Out[80]= 2^(x + y) Log[2]

I want the output to instead be 2^e1 Log[2]. I am currently using ReplaceAll as follows, but this is cumbersome in my actual application with about 20 subexpressions.

In[81]:= D[e2, x] /. e1 -> E1

Out[81]= 2^E1 Log[2]

Answer 1

Difficult to obtain and keep that form, if you set e1 to be x+y. So if you do not really need that, could instead work with replacement rules.

rul = {e1->x+y, e2->2^e1};
revrul = {x+y->e1};

InputForm[D[e2//.rul, x] /. revrul]

Out[5]//InputForm= 2^e1*Log[2]

Daniel Lichtblau Wolfram Research

Answer 2

Your answer appears to be specific due to the simple form of your e1 and e2. One possibility is to define e2 as a function in terms of e1, without specifying what e1 is:

In[8]:= Clear[e1, e2];
e2[x_] := 2^e1[x]

Then

In[10]:= D[e2[x], x]

Out[10]= 2^e1[x] Log[2] Derivative[1][e1][x]

which is a generally correct answer. As soon as you want it to compute, you can provide specific definition for e1. You can also do this inside Block, so that you don't introduce global definitions:

In[11]:= 
Block[{e1},
  e1[x_] := x + y;
  D[e2[x], x]]

Out[11]= 2^(x + y) Log[2]

I suppose this approach can scale to a larger number of sub-expressions.

HTH