# Help me with differential calculus!

I love programming and know every language, but I suck at math. Unfortunately, my school requires that computers students must take a year of calculus. There's a test next week, and I don't know any of the formulas for derivatives!

Please help me find the formulas. I need a cheat sheet - a program (as short as possible so my teacher won't notice it) that takes an expression (like 4*x^3-2) as input and outputs the derivative. (I don't care if the input and output uses command line arguments, STDIN, STDOUT, or whatever, since I'm doing all of the calculation in my head anyway.)

The test covers the following types of functions:

• Constants, like -3 or 8.5
• Power functions, like x^0.5 or x^-7
• Exponential functions, like 0.5^x or 7^x (the base is always positive)
• A constant multiplied by a function, like 3*x^5 or -0.1*0.3^x
• The sum and difference of multiple functions, like -5*x^2+10-3^x

My teacher always formats his questions in the exact same way, as shown above. He also doesn't use any fractions, numbers like pi or e, or really big numbers (bigger than 1,000). He never uses parentheses, and always shows multiplication using an asterisk (*). The only variable used is always x.

On the other hand, my teacher is pretty lenient about answers. They don't need to be simplified at all or formatted exactly as shown above, as long as it's clear what the answer is saying.

While I can use any language, remember that I can't figure out derivatives by myself. So if the program uses built-in functions to deal with equations or calculate derivatives, I won't be able to use it.

During the test, I won't have access to the Internet or any files other than the program on the cheat sheet.

Note: This scenario is entirely fictional. In real life, cheating and helping others cheat is wrong and should never be done.

• Can we expect that x is always the variable to differentiate? – Kyle Kanos Jun 17 '14 at 13:39
• Must the answer be simplified? Do we need to add like terms? – Rainbolt Jun 17 '14 at 13:40
• I guess it's time for my calculus project at scrblnrd3.github.io/Javascript-CAS to shine if I can actually golf it – scrblnrd3 Jun 17 '14 at 14:33
• Should we assume there are no parens? – Not that Charles Jun 17 '14 at 14:59
• I've answered most of these questions in my edit. There is no scientific notation or product rule. – Ypnypn Jun 17 '14 at 17:08

## Wolfram 136134 109[Thanks to Calle for his comment below]

Limited support for product and chain rules.

n=n_?NumberQ;d[v_Plus]:=d/@v;d[v_]:=v/.{x_^n:>x^(n-1)d[x]n,n^x_:>Log[n]d[x]n^x,x_*y__:>d[x]y+d[y]x,n:>0,x:>1}


Example:

d[3^(x^2)*(x^3+2*x)^2]
>> 2*3^x^2*(2+3*x^2)*(2*x+x^3) + 2*3^x^2*x*(2*x+x^3)^2*Log[3]


Note that this does not use any "built-in functions to deal with equations or calculate derivatives": only pattern-matching is involved*.

[*Well... technically the interpreter also parses and builds a sort of AST from the input too]

Ungolfed:

d[expr_Plus] := d /@ expr;
d[expr_] := expr /. {
Power[x_, n_?NumberQ] :> n Power[x, n - 1] d[x],
Power[n_?NumberQ, x_] :> Log[n] Power[n, x] d[x],
Times[x_, y__] :> d[x] y + d[y] x,
n_?NumberQ :> 0,
x :> 1
}

• This is another version. You don't have to write Power, Times etc. IDK how much that will improve your golfed version though, but you have at least one Times in there so you can def. save some characters. Also note that in your ungolfed version it says d[expr_]:= v/.... – user11030 Jun 19 '14 at 11:07
• @Calle "IDK how much that will improve your golfed version" -- 25 bytes! Cheers! – Saran Jun 19 '14 at 11:22

# Perl - 121 122

(+2 for -p)

s/(?<![-\d.*^])-?[\d.]+(?![*^\d.])/0/g;s/(?<!\^)x(?!\^)/1/g;s/x\^(-?[\d.]+)/"$1*x^".($1-1)/ge;s/([\d.]+)\^x/ln($1)*$&/g


Test:

$perl -p diff.pl << EOF > -3 > 8.5 > x^0.5 > x^-7 > 0.5^x > 7^x > 3*x^5 > -0.1*0.3^x > -5*x^2+10-3^x > EOF 0 0 0.5*x^-0.5 -7*x^-8 ln(0.5)*0.5^x ln(7)*7^x 3*5*x^4 -0.1*ln(0.3)*0.3^x -5*2*x^1+0-ln(3)*3^x  • Yet another reason for me to learn regex... – Kyle Kanos Jun 17 '14 at 15:27 • @KyleKanos Don't. Regex is bad, regex is terrific. – mniip Jun 17 '14 at 15:35 • Meh, beat me to it. Not bad! (PS: regex is beautiful) – Martin Ender Jun 17 '14 at 15:36 • I have no idea what's going on here. +1 – qwr Jun 18 '14 at 6:02 • Explanation: Constant --> 0, x --> 1, x^n --> n*x^(n-1), a^x --> ln(a)*a^x – n̴̖̋h̷͉̃a̷̭̿h̸̡̅ẗ̵̨́d̷̰̀ĥ̷̳ Jun 18 '14 at 9:56 ## Haskell 38 Chars The function d takes a function and returns a function. It is inputted in the form of a power series, and is outputted the same way (which is a type of whatever.) d=zipWith(*)[1..].tail  For example, if we input x->x^2, we get x->2*x. λ <Prelude>: d [0,0,1] [0,2]  And for the exponential function. λ <Prelude>: take 10 exp --exp redefined above to be in power series notation [1.0,1.0,0.5,0.16666666666666666,4.1666666666666664e-2,8.333333333333333e-3,1.388888888888889e-3,1.984126984126984e-4,2.48015873015873e-5,2.7557319223985893e-6] λ <Prelude>: let d=zipWith(*)[1..].tail in take 10$ d exp
[1.0,1.0,0.5,0.16666666666666666,4.1666666666666664e-2,8.333333333333333e-3,1.388888888888889e-3,1.984126984126984e-4,2.48015873015873e-5,2.7557319223985893e-6]

• But the OP doesn't know any maths! Can we expect him to express his exponential input as a power series? – Saran Jun 17 '14 at 15:52
• Well he obviously knows notation. He just doesn't know how to do the derivative operation. – PyRulez Jun 17 '14 at 15:56
• Can this handle 2^x? – Kyle Kanos Jun 17 '14 at 16:09
• What witchcraft is this? – Christofer Ohlsson Jun 18 '14 at 6:51
• I don't see where it "takes an expression (like 4*x^3-2) as input", as required by the OP. – Gabe Jun 18 '14 at 12:53

## Prolog 176

d(N,0):-number(N).
d(x,1).
d(-L,-E):-d(L,E).
d(L+R,E+F):-d(L,E),d(R,F).
d(L-R,E-F):-d(L,E),d(R,F).
d(L*R,E*R+L*F):-d(L,E),d(R,F).
d(L^R,E*R*L^(R-1)+ln(L)*F*L^R):-d(L,E),d(R,F).


Supported operators: binary +, binary -, binary *, binary ^, unary -. Note that unary + is not supported.

Sample run:

49 ?- d(-3,O).
O = 0.

50 ?- d(8.5,O).
O = 0.

51 ?- d(x^0.5,O).
O = 1*0.5*x^ (0.5-1)+ln(x)*0*x^0.5.

52 ?- d(x^-7,O).
ERROR: Syntax error: Operator expected
ERROR: d(x
ERROR: ** here **
ERROR: ^-7,O) .
52 ?- d(x^ -7,O).
O = 1* -7*x^ (-7-1)+ln(x)*0*x^ -7.

53 ?- d(x,O).
O = 1.

54 ?- d(0.5^x,O).
O = 0*x*0.5^ (x-1)+ln(0.5)*1*0.5^x.

55 ?- d(7^x,O).
O = 0*x*7^ (x-1)+ln(7)*1*7^x.

56 ?- d(3*x^5,O).
O = 0*x^5+3* (1*5*x^ (5-1)+ln(x)*0*x^5).

57 ?- d(-0.1*0.3^x,O).
O = 0*0.3^x+ -0.1* (0*x*0.3^ (x-1)+ln(0.3)*1*0.3^x).

58 ?- d(-5*x^2+10-3^x,O).
O = 0*x^2+ -5* (1*2*x^ (2-1)+ln(x)*0*x^2)+0- (0*x*3^ (x-1)+ln(3)*1*3^x).


Prolog is confused when it runs into ^- sequence. A space must be inserted between ^ and - for it to parse the expression correctly.

Hope your teacher doesn't mind the mess of equation.

Crazy time:

59 ?- d(x^x,O).
O = 1*x*x^ (x-1)+ln(x)*1*x^x.

60 ?- d((x^2-x+1)*4^ -x,O).
O = (1*2*x^ (2-1)+ln(x)*0*x^2-1+0)*4^ -x+ (x^2-x+1)* (0* -x*4^ (-x-1)+ln(4)* - 1*4^ -x).


# C, 260 248 bytes

−12 bytes thanks to ceilingcat

Hey, I think I know your teacher! Isn't it that one who has the supernatural ability to detect students executing library pattern-matching functions in their head?

So, using sscanf is out of question... But don't worry:

#define P,s--||printf(
q=94,s,c,t;main(a){char*p,i[999],*e=p=i;for(gets(i);q=c=*p++,t=q^94|c^45?c%26-16?c/46:c%16/3:1,s=(a="30PCqspP#!C@ #cS #!cpp#q"[s*5+t])/16-3,p[-1]*=~a&1,!t?0 P"*0")P"/x")P"/x*%s",e)P"*ln(%s)",e),s=0:0,e=a&2?p:e,printf(&c););}


Running examples (input on stdin; output goes to stdout):

4*x^3-2

4*x^3/x*3-2*0


This format is much better than just 12*x^2, because this way your teacher can be sure that you calculated the answer yourself and didn't cheat by copying it from someone else!

x+2^x

x/x+2^x*ln(2)


The output has a slight domain problem at x=0, but it's correct almost everywhere!

For reference, here is an ungolfed, readable (by mere mortals) version. It uses a state machine with 5 states and 5 categories of input characters.

void deriv(char* input)
{
char* p = input; // current position
char* exp = p; // base or exponent
char q = '^'; // previous character

// State machine has 5 states; here are examples of input:
// state 0: 123
// state 1: 123*
// state 2: 123*x
// state 3: 123*x^456
// state 4: 123^x
int state = 0;

// Control bits for state machine:
// bit 0: special action: stop recording base or exponent
// bit 1: special action: start recording base or exponent
// bits 4-7: if first column, specify how to calculate the derivative:
//              3 - multiply the constant term by 0
//              4 - divide x by x
//              5 - divide x^n by x and multiply by n
//              6 - multiply n^x by ln(n)
// bits 4-7: if not first column, specify the next state
//              (plus 3, to make the character printable)
const char* control =
"\x33\x30\x50\x43\x71"
"\x73\x70\x50\x23\x21"
"\x43\x40\x20\x23\x63"
"\x53\x60\x20\x23\x21"
"\x63\x70\x70\x23\x71";

for (;;) {
int c = *p++;

// Convert a char to a category:
// category 0: // - +
// category 3: // *
// category 2: // x
// category 4: // ^
// category 1: // numbers: 0...9 and decimal point
int category;
int action;

if (q == '^' && c == '-')
category = 1; // unary minus is a part of a number
else
category = c%26==16?c%16/3:c/46; // just does it

// Load new state and action to do
action = control[state * 5 + category];

if (action & 1)
p[-1] = 0;
state = (action >> 4) - 3;
if (category == 0)
{
if (state == 0)
printf("*0");
if (state == 1)
printf("/x");
if (state == 2)
printf("/x*%s", exp);
if (state == 3)
printf("*ln(%s)", exp);
state = 0;
}
if (action & 2)
exp = p;

if (c == 0 || c == '\n') // either of these can mark end of input
break;

putchar(c);
q = c;
}
}


P.S. Watch out for that gets function: it has a security vulnerability that can let your teacher execute a rootkit in your mind by providing too long input...

• @ceilingcat Fail on -x^-1 what does *0-x^/x*-1*0 mean? – l4m2 Jun 5 '20 at 7:12
• Oh you assume input 0-x^-1 but 0*0-x^/x*-1*0 still has a ^/ – l4m2 Jun 5 '20 at 7:13

# Lua 296268 263

function d(a)l=""i=a:find"x" if i then if a:sub(i-1,i-1)=="^"then l="log("..a:sub(1,i-2)..")*"..a elseif a:sub(i+1,i+1)=="^"then l=a:sub(i+2).."*"..a:sub(1,i)p=a:sub(i+2)-1 if p~=1 then l= l..a:sub(i+1,i+1)..p end else l=a:sub(1,i-2)end else l="0"end return l end


Not very golfed and cannot currently handle multiple terms (you can just run it a few times, right?), but it can handle n^x, x^n and n as input.

Ungolfed...

function d(a)
l=""
i=a:find"x"
if i then
if a:sub(i-1,i-1)=="^" then
l="log("..a:sub(1,i-2)..")*"..a
elseif a:sub(i+1,i+1)=="^" then
l=a:sub(i+2).."*"..a:sub(1,i)
p=a:sub(i+2)-1 -- this actually does math here
if p~=1 then
l= l..a:sub(i+1,i+1)..p
end
else
l=a:sub(1,i-2)
end
else
l="0"
end
return l
end

• str.func(str,...) == str:func(...), that's why strings got the metatable after all... – mniip Jun 17 '14 at 14:36
• @mniip: Still learning Lua. Thanks for the tip. – Kyle Kanos Jun 17 '14 at 14:38
• Since the OP is only looking for code "he can calculate in his head", I wouldn't bother with defining a function and declaring l local. Just expect the input to be stored in a and say the output will be stored in l. – Martin Ender Jun 17 '14 at 14:44
• You can omit parentheses in a:find("x"), also note that 1then only works in Lua 5.2 – mniip Jun 17 '14 at 14:47
• @mniip: Whoa, that's pretty cool that () is optional. The 1then was just fixed as I don't have 5.2 (not doing any CPU updates until after dissertation is done b/c I don't want to mess anything up). – Kyle Kanos Jun 17 '14 at 14:49

## ECMAScript 6, 127 bytes

Here is my regex attempt (using a single regex and some logic in the replacement callback):

i.replace(/(^|[*+-])(\d+|(?:([\d.]+)\^)?(x)(?:\^(-?[\d.]+))?)(?![.*^])/g,(m,s,a,b,x,e)=>s+(b?'ln'+b+'*'+a:e?e--+'*x^'+e:x?1:0))


This expects the input string to be stored in i and simply returns the result. Try it out in an ECMAScript 6 compliant console (like Firefox's).

## sed, 110

Taking very literally "They don't need to be simplified at all or formatted exactly as shown above, as long as it's clear what the answer is saying":

s/.*/__&_/;s/x\^(-?[0-9.]+)/\1*x^(\1-1)/g;s/([0-9.]+)\^/ln\1*\1^/g;s/([^(][-+_])[0-9.]+([-+_])/\10\2/g;s/_//g


The byte count includes 1 for the r flag.

# Add underscores before and after the string, to help with solo-constant recognition
s/.*/__&_/
# Power rule: replace x^c with c*x^(c-1) where c is a number
s/x\^(-?[0-9.]+)/\1*x^(\1-1)/g
# Exponentials: replace c^ with lnc*c^ where c is a number
# (This assumes that there will be an x after the ^)
s/([0-9.]+)\^/ln\1*\1^/g
# Constants: replace ?c? with ?0? where c is a number and ? is +, -, or _
# Except if it's prededed by a parenthesis then don't, because this matches c*x^(c-1)!
s/([^(][-+_])[0-9.]+([-+_])/\10\2/g
# Get rid of the underscores
s/_//g


Sample run:

$cat derivatives.txt -3 8.5 x^0.5 x^-7 0.5^x 7^x 3*x^5 -0.1*0.3^x -5*x^2+10-3^x$ sed -re 's/.*/__&_/;s/x\^(-?[0-9.]+)/\1*x^(\1-1)/g;s/([0-9.]+)\^/ln\1*\1^/g;s/([^(][-+_])[0-9.]+([-+_])/\10\2/g;s/_//g' derivatives.txt
-0
0
0.5*x^(0.5-1)
-7*x^(-7-1)
ln0.5*0.5^x
ln7*7^x
3*5*x^(5-1)
-0.1*ln0.3*0.3^x
-5*2*x^(2-1)+0-ln3*3^x


I bet this could be golfed further; it's my first try at sed. Fun!

## Ruby, 152

...or 150 if you don't need to print... or 147 if you also are ok with an array that you need to join yourself.

run with ruby -nal

p gsub(/(?<!\^)([-+])/,'#\1').split(?#).map{|s|s[/x\^/]?$+$'+"x^(#{$'}-1)":s[/-?(.*)\^(.*)x/]?s+"*ln(#{$1}*#{$2[0]?$2:1})":s[/\*?x/]?($[0]?$:1):p}*''


ungolfed:

p gsub(/(?<!\^)([-+])/,'#\1').split(?#). # insert a # between each additive piece, and then split.
map{ |s|
if s[/x\^/]                          # if it's c*x^a
$ +$' + "x^(#{$'}-1)" # return c*ax^(a-1) elsif s[/-?(.*)\^(.*)x/] # if it's c*b^(a*x) ln =$1 + ?* + ($2[0] ?$2 : 1)  #      return c*b^(a*x)*ln(b*a)
s+"*ln(#{ln})"
elsif s[/\*?x/]                      # if it's c*x
($[0] ?$ : 1)                 #      return c
else                                 # else (constant)
nil                              #      return nil
end
}*''


My main problem with this one is the number of characters proper splitting takes. The only other way I could think of was split(/(?<!\^)([-+])/) which gives + and - as their own results. Any hints for a better solution?

Also, is there a shorter way to return s if it's not empty, but otherwise return y? I've used s[0]?y:s? In JS I'd just do s||y, but "" is truthy in Ruby.

• Would a lookahead assertion help, like so: split(/(?<!\^)(?=[-+])/)`? – DLosc Jun 19 '14 at 0:43