# Defining a function that differentiates another function with rule replacement

This question is related to the question I posted here. In that question, I thought I have simplified my original problem into an equivalent concise version, but I found that it is not. So I decided to open this question.

The goal is the same that I want to define a function hhpar that will differentiate hh with respect to x, while I changed the rule function as follows:

Clear[rule];
rulelist = {{tt -> 1, zz -> 1}, {tt -> 1, zz -> 2}, {tt -> 2,
zz -> 1}, {tt -> 2, zz -> 2}};
rule[i_] := rulelist[[i]];


The hh function is

Clear[hh];
hh[x_, y_, i_] := {{tt, x + Sin[x]},{y, zz}}/.rule[i];


As suggested by Szabolcs, to defined hhpar as

hhpar[x_,y_,i_]:=Derivative[1, 0, 0][hh][x, y, i]


will solve my former question. But it didn't work in current problem because it only gives

Using Trace, I figure out that the problem is due to the failure of rule replacement before differentiation. But how would I define hhpar now to solve the problem?

• Try with Derivative[1, 0, 0][hh[x, y, i]]
– eldo
Commented Jul 6, 2014 at 11:54
• @eldo Thank you, eldo! But it doesn't work. Commented Jul 6, 2014 at 12:29
• define it this way: hhpar [x_, y_, i_] := Derivative[1, 0][Evaluate@hh[#1, #2, i] &][x, y] Commented Jul 7, 2014 at 21:03

## Edit

This edited answer reflects comments made on my original version by the OP.

Clear[rulelist, rule];
rulelist = {{tt -> 1, zz -> 1}, {tt -> 1, zz -> 2}, {tt -> 2, zz -> 1}, {tt -> 2, zz -> 2}};
Table[rule[i] = rulelist[[i]], {i, 4}];


To see where you went wrong let's look at what Derivative returns when it is given your version of hh.

Clear[hh];
hh[x_, y_, i_] := {{tt, x + Sin[x]}, {y, zz}} /. rule[i]
Derivative[1, 0, 0][hh]


ReplaceAll::reps: {rule[#3]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>

D[{{tt, Sin[#1] + #1}, {#2, zz}} /. rule[#3], #1] &


This tells us quite a lot about what is going wrong.

Derivative is trying to take the 1st partial derivative of the 1st independent variable of a function of three variables, and that function is a ReplaceAll expression. There is no way that can work, and it isn't at all what you want anyway. In this case, you want Derivative to return a function that is the 1st partial derivative of the 1st independent variable of a function of two variables.

Clearly hh needs to be redefined to be a function of one variable, the rule index, returning a form which is a function of two variables and where all the desired replacements have been made. While we are at it, we may as well make hh return a pure function. This will avoid any possible scoping issues. We will also generalize hhpar so it can be used to produce any of the 1st order partials of hh[i]. It also turns out that we get the 0 order partial, h[i] itself, for free (hh[i] == hpar[0, 0][#1, #2, i] &).

Clear[hh, hhpar];
hh[i_] = {{tt, #1 + Sin[#1]}, {#2, zz}} & /. rule[i];
hhpar[j_, k_][x_, y_, i_] := Derivative[j, k][hh[i]][x, y]


Now we have the situation where

hh[1]

{{1, #1 + Sin[#1]}, {#2, 1}} &


and

Derivative[1, 0][hh[1]]

{{0, 1 + Cos[#1]}, {0, 0}} &


exactly as expected.

Computing the 1st partial derivatives gives for all the rules

Table[hhpar[1, 0][x, y, i], {i, 4}]


gives

{{{0, 1 + Cos[x]}, {0, 0}}, {{0, 1 + Cos[x]}, {0, 0}},
{{0, 1 + Cos[x]}, {0, 0}}, {{0, 1 + Cos[x]}, {0, 0}}}

Table[hhpar[0, 1][x, y, i], {i, 4}]

{{{0, 0}, {1, 0}}, {{0, 0}, {1, 0}}, {{0, 0}, {1, 0}}, {{0, 0}, {1, 0}}}

• Thank you, m_goldberg. Your definition really works, but I don't understand why my definition gives wrong answer. Could you explain a little? I think I miss a lesson on function definition. Commented Jul 7, 2014 at 12:13
• And I forgot what this kind of function definition hhpar[j_, k_][x_, y_, i_] is called? Could you show me a link which has a discussion on this kind of function definition? Commented Jul 7, 2014 at 12:19
• @matheorem. I don't have time now to update my answer to explain how your code goes wrong. I will do so latter today. As for a link on non-symbolic heads, look at this Commented Jul 7, 2014 at 14:03
• I found this works: hh[i_][x_, y_] := {{tt, x + Sin[x]}, {y, 1}} /. rulelist[[i]]; hhpar[x_, y_, i_] := Derivative[1, 0, 0][hh[i][#1, #2] &][x, y]; While this not works: hh[x_, y_, i_] := {{tt, x + Sin[x]}, {y, 1}} /. rulelist[[i]]; hhpar[x_, y_, i_] := Derivative[1, 0, 0][hh[#1, #2, i] &][x, y]; Commented Jul 7, 2014 at 14:57
• Nice answer. +1 Commented Jul 8, 2014 at 0:41