# Definition

The chain rule with two functions state that:

D[f(g(x))] = f'(g(x)) * g'(x)


Or, alternatively:

D[f1(f2(x))] = f1'(f2(x)) * f2'(x)


The chain rule with three functions state that:

D[f(g(h(x)))] = f'(g(h(x))) * g'(h(x)) * h'(x)


Or, alternatively:

D[f1(f2(f3(x)))] = f1'(f2(f3(x))) * f2'(f3(x)) * f3'(x)


Et cetera.

• Given an integer between 2 and 21, output the chain rule with that many functions, either in the first form or in the second form.
• Please specify if you are using the second form.

# Specs

• The format of the string must be exactly that stated above, with:
1. all the spaces kept intact
2. a capitalized D
3. a square bracket immediately following D
4. the asterisk kept intact
• One extra trailing space (U+0020) is allowed.
• Leading zeros in the function names in the second form (e.g. f01 instead of f1) is allowed.

# Testcases

If you use the first form:

input output
2     D[f(g(x))] = f'(g(x)) * g'(x)
3     D[f(g(h(x)))] = f'(g(h(x))) * g'(h(x)) * h'(x)


If you use the second form:

input output
2     D[f1(f2(x))] = f1'(f2(x)) * f2'(x)
3     D[f1(f2(f3(x)))] = f1'(f2(f3(x))) * f2'(f3(x)) * f3'(x)


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body{text-align:left!important}#answer-list,#language-list{padding:10px;width:290px;float:left}table thead{font-weight:700}table td{padding:5px}
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script> <link rel="stylesheet" type="text/css" href="//cdn.sstatic.net/codegolf/all.css?v=83c949450c8b"> <div id="answer-list"> <h2>Leaderboard</h2> <table class="answer-list"> <thead> <tr><td></td><td>Author</td><td>Language</td><td>Size</td></tr></thead> <tbody id="answers"> </tbody> </table> </div><div id="language-list"> <h2>Winners by Language</h2> <table class="language-list"> <thead> <tr><td>Language</td><td>User</td><td>Score</td></tr></thead> <tbody id="languages"> </tbody> </table> </div><table style="display: none"> <tbody id="answer-template"> <tr><td>{{PLACE}}</td><td>{{NAME}}</td><td>{{LANGUAGE}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr></tbody> </table> <table style="display: none"> <tbody id="language-template"> <tr><td>{{LANGUAGE}}</td><td>{{NAME}}</td><td>{{SIZE}}</td><td><a href="{{LINK}}">Link</a></td></tr></tbody> </table>

• Do function names have to be lowercase? Jul 26, 2016 at 23:21
• @betseg Yes of course. Jul 26, 2016 at 23:22

## Python 2, 79 bytes

f=lambda n:0**n*"D[x] ="or f(n-1).replace("x","f%d(x)"%n)+1%n*" *"+" f%d'(x)"%n


Outputs with numbered functions.

Builds the output by repeatedly replacing each x with fn(x), then appending * fn'(x). The * is omitted for n==1.

Compare to the iterate program (92 bytes):

r="D[x] = ";n=0
exec'n+=1;r=r.replace("x","f%d(x)"%n)+"f%d\'(x) * "%n;'*input()
print r[:-3]


96 bytes:

n=input();r='';s='x'
while n:s='f%d(%%s)'%n%s;r=" * f%d'"%n+s[2:]+r;n-=1
print"D[%s] = "%s+r[3:]


Outputs with numbered functions.

Accumulates the nested function f1(f2(f3(x))) in s and the right-hand-side expression in r. The string formatting is clunky; f-strings from 3.6 would do better.

# Sesos, 49 bytes

0000000: 2ac992 63fb92 245fb6 6c57be 255bbe 2cc9bf 6d49da  *..c..$_.lW.%[.,..mI. 0000015: 025e7f fdced0 fd67f8 fcde33 b6a7b2 643d4f 65597e .^.....g...3...d=OeY~ 000002a: f77a72 dd73cf fe .zr.s..  Try it online ### Disassembled set numin add 68 ; 'D' put sub 7 ; '=' - 'D' fwd 1 add 32 ; ' ' fwd 1 add 91 ; '[' put add 2 ; ']' - '[' fwd 1 add 102 ; 'f' fwd 1 add 40 ; '(' fwd 3 add 120 ; 'x' rwd 2 get jmp jmp fwd 1 add 1 rwd 3 put add 1 fwd 1 put fwd 1 sub 1 jnz sub 1 rwd 1 add 1 ; ')' - '(' fwd 3 put rwd 1 jmp rwd 3 sub 1 fwd 1 put fwd 1 add 1 fwd 1 sub 1 jnz rwd 4 put get add 41 ; ')' rwd 1 put rwd 1 put get add 42 ; '*' fwd 1 put fwd 2 put add 1 fwd 1 sub 2 ; '\'' - ')' put add 1 ; '(' - '\'' put fwd 1 jnz fwd 2 put rwd 5 put  # JavaScript (ES6), 89 bytes f=n=>--n?f(n).replace(/x/g,${c=(n+15).toString(36)}(x))+ * ${c}'(x):D[f(x)] = f'(x)  Or 75 bytes using the second form: f=n=>n>1?f(n-1).replace(/x/g,f${n}(x))+ * f${n}'(x):D[f1(x)] = f1'(x)  Or 82/64 bytes if I'm allowed an extra 1 * term: f=n=>n?f(n-1).replace(/x/g,${c=(n+14).toString(36)}(x))+ * ${c}'(x):D[x] = 1 f=n=>n?f(n-1).replace(/x/g,f${n}(x))+ * f${n}'(x):D[x] = 1  • Why don't you put the 73-byte version as your main? Jul 26, 2016 at 23:02 • If you're being recursive, you need the f= or else your program won't work. Jul 26, 2016 at 23:21 • @KevinLau-notKenny Bah, I always forget to do that. – Neil Jul 27, 2016 at 0:09 # Ruby, 72 bytes Second form. Port to Ruby from @Neil answer. f=->n{n>1?f[n-1].gsub(?x,"f#{n}(x)")+" * f#{n}'(x)":"D[f1(x)] = f1'(x)"}  # Julia, 66 bytes !x=x>1?replace(!~-x,"x","f$x(x)")*" * f\$x'(x)":"D[f1(x)] = f1'(x)"


Port of @Neil's ES6 answer. Uses the second form.

# Python 2, 174 bytes

i=input()+1
a=lambda x:'('.join(['f%d'%e for e in range(x,i)])+'(x'+')'*(i-x)
print'D[%s] = %s'%(a(1),' * '.join(["f%d'%s%s)"%(e+1,'('*(i-e-2>0),a(e+2))for e in range(i-1)]))


Lots of room left to be golfed here. Uses the second form (f1, f2, etc.).

# JavaScript (ES6), 194 bytes

n=>{s="D[";for(i=1;i<=n;i++)s+="f"+i+"(";s+="x";s+=")".repeat(n);s+="] = ";for(i=1;i<=n;i++){s+="f"+i+"'(";for(j=i+1;j<=n;j++)s+="f"+j+"(";s+="x";s+=")".repeat(n-i+1);if(i-n)s+=" * ";}return s;}


31 bytes saved thanks to @LeakyNun.

It's an anonymous lambda function. I'm sure there's some way to shorten this...

## Ungolfed

var chain = function(n) {
var str = "D["; // derivative symbol
for (var i = 1; i <= n; i++) {
str += "f"+i+"("; // put in n functions, each labeled f1(, f2(, etc.
}
str += "x"; // add in the function input, usually "x"
for (var i = 0; i < n; i++) {
str += ")"; // add in n end parentheses
}
str += "] = "; // add in the end bracket and the equals operator
for (var i = 1; i <= n; i++) {
str += "f"+i+"'("; // add in all n of the outer functions with the prime operator
for (var j = i+1; j <= n; j++) {
str += "f"+j+"("; // add in all of the inner functions
}
str += "x"; // add in the input, "x"
for (var j = 1; j <= n; j++) {
str += ")"; // close the parentheses
}
if (i !== n) {
str += " * "; // the multiplication of all of the outer primed functions
}
}
return str;
};

• Change str into s. Jul 26, 2016 at 23:25
• Ok, I forgot to change it. Lol Jul 26, 2016 at 23:27
• Change for(j=i;j<=n;j++)s+=")"; to s+=")".repeat(n-i+1); Jul 27, 2016 at 0:30