Polynomial derivative

Polynomial derivative is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Given a polynomial, represented by an ordered list of its coefficients by increasing degree (e.g. [-1, 6, 5] represents 5x²+6x-1), calculate the polynomial representing the derivative. For example, the derivative of the aforementioned polynomial is 10x+6, represented by [6, 10]. Test cases: 5, -3x+4, 5x²+6x-1, x³-2x²+3x-4, -x⁴-x³+x+1

Related task

•   Polynomial long division

ALGOL 68

<lang algol68>BEGIN # find the derivatives of polynominals, given their coefficients #

   # returns the derivative polynominal of the polynominal defined by #
   #         the array of coeficients, where the coefficients are in  #
   #         order of ioncreasing power of x                          #
   OP DERIVATIVE = ( []INT p )[]INT:
   BEGIN
       [ 1 : UPB p - 1 ]INT result;
       FOR i FROM 2 TO UPB p DO
           result[ i - 1 ] := ( i - 1 ) * p[ i ]
       OD;
       result
   END # DERIVATIVE # ;
   # prints the polynomial defined by the coefficients in p #
   OP SHOW = ( []INT p )VOID:
   BEGIN
       BOOL first := TRUE;
       FOR i FROM UPB p BY -1 TO LWB p DO
           IF p[ i ] /= 0 THEN
               IF first THEN
                   IF   p[ i ] < 0 THEN print( ( "-" ) ) FI
               ELSE
                   IF   p[ i ] < 0
                   THEN print( ( " - " ) )
                   ELSE print( ( " + " ) )
                   FI
               FI;
               first := FALSE;
               IF   i = LWB p
               THEN print( ( whole( ABS p[ i ], 0 ) ) )
               ELSE
                   IF ABS p[ i ] > 1 THEN print( ( whole( ABS p[ i ], 0 ) ) ) FI;
                   print( ( "x" ) );
                   IF i > LWB p + 1 THEN print( ( "^", whole( i - 1, 0 ) ) ) FI
               FI
           FI
       OD;
       IF first THEN
           # all coefficients were 0 #
           print( ( "0" ) )
       FI
   END # SHOW # ;
   # task test cases #
   PROC test = ( []INT p )VOID: BEGIN SHOW p; print( ( " -> " ) ); SHOW DERIVATIVE p; print( ( newline ) ) END;
   test( ( 5 ) ); test( ( 4, -3 ) ); test( ( -1, 6, 5 ) ); test( ( -4, 3, -2, 1 ) ); test( ( 1, 1, 0, -1, -1 ) )

END</lang>

Output:

5 -> 0
-3x + 4 -> -3
5x^2 + 6x - 1 -> 10x + 6
x^3 - 2x^2 + 3x - 4 -> 3x^2 - 4x + 3
-x^4 - x^3 + x + 1 -> -4x^3 - 3x^2 + 1

Factor

<lang factor>USING: generalizations kernel math.polynomials prettyprint ;

{ 5 } { 4 -3 } { -1 6 5 } { -4 3 -2 1 } { 1 1 0 -1 -1 }

[ pdiff ] 5 napply .s clear</lang>

Output:

{ }
{ -3 }
{ 6 10 }
{ 3 -4 3 }
{ 1 0 -3 -4 }

The implementation of pdiff:

<lang factor>USING: kernel math.vectors sequences ; IN: math.polynomials

pdiff ( p -- p' ) dup length <iota> v* rest ;</lang>

FreeBASIC

<lang freebasic>sub polydiff( p() as integer )

   'differentiates the polynomial
   'p(0) + p(1)x + p(2)x^2 +... + p(n)x^n
   'in place
   dim as integer i, n = ubound(p)
   if n=0 then
       p(0)=0
       return
   end if
   for i = 0 to n - 1
       p(i) = (i+1)*p(i+1)
   next i
   redim preserve p(0 to n-1)
   return

end sub

sub print_poly( p() as integer )

   'quick and dirty display of the poly
   if ubound(p)=0 and p(0)=0 then
       print 0
       return
   end if
   for i as integer = 0 to ubound(p)
       if i = 0 then print p(i);" ";
       if i = 1 and p(i)>0 then print using "+ #x";p(i);
       if i = 1 and p(i)<0 then print using "- #x";-p(i);
       if i > 1 and p(i)>0 then print using "+ #x^#";p(i);i;
       if i > 1 and p(i)<0 then print using "- #x^#";-p(i);i;        
   next i
   print

end sub

'test cases redim as integer p(0) p(0) = 5 print_poly(p()) print "Differentiates to " polydiff(p()) print_poly(p()): print

redim as integer p(1) p(0) = 4 : p(1) = -3 print_poly(p()) print "Differentiates to " polydiff(p()) print_poly(p()): print

redim as integer p(2) p(0) = -1 : p(1) = 6 : p(2) = 5 print_poly(p()) print "Differentiates to " polydiff(p()) print_poly(p()): print

redim as integer p(3) p(0) = 4 : p(1) = 3 : p(2) = -2 : p(3) = 1 print_poly(p()) print "Differentiates to " polydiff(p()) print_poly(p()): print

redim as integer p(4) p(0) = 1 : p(1) = 1 : p(2) = 0 : p(3) = -1 : p(4) = -1 print_poly(p()) print "Differentiates to " polydiff(p()) print_poly(p()): print</lang>

Output:

5 
Differentiates to 
0
4 - 3x
Differentiates to 
-3 
-1 + 6x+ 5x^2
Differentiates to 
6 + %10x
4 + 3x- 2x^2+ 1x^3
Differentiates to 
3 - 4x+ 3x^2
1 + 1x- 1x^3- 1x^4
Differentiates to 

1 - 3x^2- 4x^3

Go

Translation of: Wren

<lang go>package main

import (

   "fmt"
   "strings"

)

func derivative(p []int) []int {

   if len(p) == 1 {
       return []int{0}
   }
   d := make([]int, len(p)-1)
   copy(d, p[1:])
   for i := 0; i < len(d); i++ {
       d[i] = p[i+1] * (i + 1)
   }
   return d

}

var ss = []string{"", "", "\u00b2", "\u00b3", "\u2074", "\u2075", "\u2076", "\u2077", "\u2078", "\u2079"}

// for n <= 20 func superscript(n int) string {

   if n < 10 {
       return ss[n]
   }
   if n < 20 {
       return ss[1] + ss[n-10]
   }
   return ss[2] + ss[0]

}

func abs(n int) int {

   if n < 0 {
       return -n
   }
   return n

}

func polyPrint(p []int) string {

   if len(p) == 1 {
       return fmt.Sprintf("%d", p[0])
   }
   var terms []string
   for i := 0; i < len(p); i++ {
       if p[i] == 0 {
           continue
       }
       c := fmt.Sprintf("%d", p[i])
       if i > 0 && abs(p[i]) == 1 {
           c = ""
           if p[i] != 1 {
               c = "-"
           }
       }
       x := "x"
       if i <= 0 {
           x = ""
       }
       terms = append(terms, fmt.Sprintf("%s%s%s", c, x, superscript(i)))
   }
   for i, j := 0, len(terms)-1; i < j; i, j = i+1, j-1 {
       terms[i], terms[j] = terms[j], terms[i]
   }
   s := strings.Join(terms, "+")
   return strings.Replace(s, "+-", "-", -1)

}

func main() {

   fmt.Println("The derivatives of the following polynomials are:\n")
   polys := [][]int{{5}, {4, -3}, {-1, 6, 5}, {-4, 3, -2, 1}, {1, 1, 0, -1, -1}}
   for _, poly := range polys {
       deriv := derivative(poly)
       fmt.Printf("%v -> %v\n", poly, deriv)
   }
   fmt.Println("\nOr in normal mathematical notation:\n")
   for _, poly := range polys {
       deriv := derivative(poly)
       fmt.Println("Polynomial : ", polyPrint(poly))
       fmt.Println("Derivative : ", polyPrint(deriv), "\n")
   }

}</lang>

Output:

The derivatives of the following polynomials are:

[5] -> [0]
[4 -3] -> [-3]
[-1 6 5] -> [6 10]
[-4 3 -2 1] -> [3 -4 3]
[1 1 0 -1 -1] -> [1 0 -3 -4]

Or in normal mathematical notation:

Polynomial :  5
Derivative :  0 

Polynomial :  -3x+4
Derivative :  -3 

Polynomial :  5x²+6x-1
Derivative :  10x+6 

Polynomial :  x³-2x²+3x-4
Derivative :  3x²-4x+3 

Polynomial :  -x⁴-x³+x+1
Derivative :  -4x³-3x²+1 

Haskell

<lang haskell>deriv = zipWith (*) [1..] . tail

main = mapM_ (putStrLn . line) ps

 where
   line p = "\np  = " ++ show p ++ "\np' = " ++ show (deriv p)
   ps = [[5],[4,-3],[-1,6,5],[-4,3,-2,1],[1,1,0,-1,-1]]</lang>

main

p  = [5]
p' = []

p  = [4,-3]
p' = [-3]

p  = [-1,6,5]
p' = [6,10]

p  = [-4,3,-2,1]
p' = [3,-4,3]

p  = [1,1,0,-1,-1]
p' = [1,0,-3,-4]

With fancy output

<lang haskell>{-# language LambdaCase #-}

showPoly [] = "0" showPoly p = foldl1 (\r -> (r ++) . term) $

            dropWhile null $
            foldMap (\(c, n) -> [show c ++ expt n]) $
            zip p [0..]
 where
   expt = \case 0 -> ""
                1 -> "*x"
                n -> "*x^" ++ show n

   term = \case [] -> ""
                '0':'*':t -> ""
                '-':'1':'*':t -> " - " ++ t
                '1':'*':t -> " + " ++ t
                '-':t -> " - " ++ t
                t -> " + " ++ t

main = mapM_ (putStrLn . line) ps

 where
   line p = "\np  = " ++ showPoly p ++ "\np' = " ++ showPoly (deriv p)
   ps = [[5],[4,-3],[-1,6,5],[-4,3,-2,1],[1,1,0,-1,-1]]</lang>

 main

p  = 5
p' = 0

p  = 4 - 3*x
p' = -3

p  = -1 + 6*x + 5*x^2
p' = 6 + 10*x

p  = -4 + 3*x - 2*x^2 + x^3
p' = 3 - 4*x + 3*x^2

p  = 1 + 1*x - 1*x^3 - 1*x^4
p' = 1 - 3*x^2 - 4*x^

jq

Adapted from Wren (with corrections)

Works with: jq

Works with gojq, the Go implementation of jq

The following definition of polyPrint has no restriction on the degree of the polynomial.

<lang jq># The input should be a non-empty array of integers representing a polynomial.

1. The output likewise represents its derivative.

def derivative:

 . as $p
 | if length == 1 then [0]
   else reduce range(0; length-1) as $i (.[1:];
     .[$i] = $p[$i+1] * ($i + 1) )
   end;

def polyPrint:

 def ss: ["\u2070", "\u00b9", "\u00b2", "\u00b3", "\u2074", "\u2075", "\u2076", "\u2077", "\u2078", "\u2079"];
 def digits: tostring | explode[] | [.] | implode | tonumber;
 ss as $ss 
 | def superscript:
     if . <= 1 then ""
     else reduce digits as $d (""; . + $ss[$d] )
     end;

 . as $p
 | if length == 1 then .[0] | tostring
   else reduce range(0; length) as $i ([];
       if $p[$i] != 0

then (if $i > 0 then "x" else "" end) as $x

       | ( if $i > 0 and ($p[$i]|length) == 1

then (if $p[$i] == 1 then "" else "-" end) else ($p[$i]|tostring) end ) as $c | . + ["\($c)\($x)\($i|superscript)"]

       else . end )
   | reverse
   | join("+")
   | gsub("\\+-"; "-")
   end ;

def task:

 def polys: [ [5], [4, -3], [-1, 6, 5], [-4, 3, -2, 1], [1, 1, 0, -1, -1] ]; 

 "Example polynomials and their derivatives:\n",
 ( polys[] |  "\(.) -> \(derivative)" ),

 "\nOr in normal mathematical notation:\n",
 ( polys[]
   | "Polynomial : \(polyPrint)",
     "Derivative : \(derivative|polyPrint)\n" ) ;

task</lang>

Output:

Example polynomials and their derivatives:

[5] -> [0]
[4,-3] -> [-3]
[-1,6,5] -> [6,10]
[-4,3,-2,1] -> [3,-4,3]
[1,1,0,-1,-1] -> [1,0,-3,-4]

Or in normal mathematical notation:

Polynomial : 5
Derivative : 0

Polynomial : -3x+4
Derivative : -3

Polynomial : 5x²+6x-1
Derivative : 10x+6

Polynomial : x³-2x²+3x-4
Derivative : 3x²-4x+3

Polynomial : -x⁴-x³+x+1
Derivative : -4x³-3x²+1

Julia

<lang julia>using Polynomials

testcases = [

   ("5", [5]),
   ("-3x+4", [4, -3]),
   ("5x2+6x-1", [-1, 6, 5]),
   ("x3-2x2+3x-4", [-4, 3, -2, 1]),
   ("-x4-x3+x+1", [1, 1, 0, -1, -1]),

]

for (s, coef) in testcases

   println("Derivative of $s: ", derivative(Polynomial(coef)))

end

</lang>

Output:

Derivative of 5: 0
Derivative of -3x+4: -3
Derivative of 5x2+6x-1: 6 + 10*x
Derivative of x3-2x2+3x-4: 3 - 4*x + 3*x^2
Derivative of -x4-x3+x+1: 1 - 3*x^2 - 4*x^3

Perl

<lang perl>use strict; use warnings; use feature 'say'; use utf8; binmode(STDOUT, ':utf8');

sub pp {

   my(@p) = @_;
   return 0 unless @p;
   my @f = $p[0];
   push @f, ($p[$_] != 1 and $p[$_]) . 'x' . ($_ != 1 and (qw<⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹>)[$_])
       for grep { $p[$_] != 0 } 1 .. $#p;
   ( join('+', reverse @f) =~ s/-1x/-x/gr ) =~ s/\+-/-/gr

}

for ([5], [4,-3], [-1,3,-2,1], [-1,6,5], [1,1,0,-1,-1]) {

   my @poly = @$_;
   say 'Polynomial: ' . join(', ', @poly) . ' ==> ' . pp @poly;
   $poly[$_] *= $_ for 0 .. $#poly;
   shift @poly;
   say 'Derivative: ' . (@poly ? join', ', @poly : 0) . ' ==> ' . pp(@poly) . "\n";

}</lang>

Output:

Polynomial: 5 ==> 5
Derivative: 0 ==> 0

Polynomial: 4, -3 ==> -3x+4
Derivative: -3 ==> -3

Polynomial: -1, 3, -2, 1 ==> x³-2x²+3x-1
Derivative: 3, -4, 3 ==> 3x²-4x+3

Polynomial: -1, 6, 5 ==> 5x²+6x-1
Derivative: 6, 10 ==> 10x+6

Polynomial: 1, 1, 0, -1, -1 ==> -x⁴-x³+x+1
Derivative: 1, 0, -3, -4 ==> -4x³-3x²+1

Phix

--
-- demo\rosetta\Polynomial_derivative.exw
--
with javascript_semantics
function derivative(sequence p)
    if p={} then return {} end if
    sequence r = repeat(0,length(p)-1)
    for i=1 to length(r) do
        r[i] = i*p[i+1]
    end for
    return r
end function

function poly(sequence si)
-- display helper, copied from demo\rosetta\Polynomial_long_division.exw
    string r = ""
    for t=length(si) to 1 by -1 do
        integer sit = si[t]
        if sit!=0 then
            if sit=1 and t>1 then
                r &= iff(r=""? "":" + ")
            elsif sit=-1 and t>1 then
                r &= iff(r=""?"-":" - ")
            else
                if r!="" then
                    r &= iff(sit<0?" - ":" + ")
                    sit = abs(sit)
                end if
                r &= sprintf("%d",sit)
            end if
            r &= iff(t>1?"x"&iff(t>2?sprintf("^%d",t-1):""):"")
        end if
    end for
    if r="" then r="0" end if
    return r
end function

constant tests = {{5},{4,-3},{-1,6,5},{-4,3,-2,1},{1,1,0,-1,-1}}
for i=1 to length(tests) do
    sequence t = tests[i],
             r = derivative(t)
    printf(1,"%20s ==> %16s   (internally %v -> %v)\n",{poly(t),poly(r),t,r})
end for

?"done"
{} = wait_key()

Output:

                   5 ==>                0   (internally {5} -> {})
             -3x + 4 ==>               -3   (internally {4,-3} -> {-3})
       5x^2 + 6x - 1 ==>          10x + 6   (internally {-1,6,5} -> {6,10})
 x^3 - 2x^2 + 3x - 4 ==>    3x^2 - 4x + 3   (internally {-4,3,-2,1} -> {3,-4,3})
  -x^4 - x^3 + x + 1 ==> -4x^3 - 3x^2 + 1   (internally {1,1,0,-1,-1} -> {1,0,-3,-4})

Raku

<lang perl6>use Lingua::EN::Numbers:ver<2.8+>;

sub pretty (@poly) {

   join( '+', (^@poly).reverse.map: { @poly[$_] ~ "x{.&super}" } )\
   .subst(/['+'|'-']'0x'<[⁰¹²³⁴⁵⁶⁷⁸⁹]>*/, , :g).subst(/'x¹'<?before <-[⁰¹²³⁴⁵⁶⁷⁸⁹]>>/, 'x')\
   .subst(/'x⁰'$/, ).subst(/'+-'/, '-', :g).subst(/(['+'|'-'|^])'1x'/, {"$0x"}, :g) || 0

}

for [5], [4,-3], [-1,3,-2,1], [-1,6,5], [1,1,0,-1,-1] -> $test {

  say "Polynomial: " ~ "[{$test.join: ','}] ➡ " ~ pretty $test;
  my @poly = |$test;
  (^@poly).map: { @poly[$_] *= $_ };
  shift @poly;
  say "Derivative: " ~ "[{@poly.join: ','}] ➡ " ~ pretty @poly;
  say ;

}</lang>

Output:

Polynomial: [5] ➡ 5
Derivative: [] ➡ 0

Polynomial: [4,-3] ➡ -3x+4
Derivative: [-3] ➡ -3

Polynomial: [-1,3,-2,1] ➡ x³-2x²+3x-1
Derivative: [3,-4,3] ➡ 3x²-4x+3

Polynomial: [-1,6,5] ➡ 5x²+6x-1
Derivative: [6,10] ➡ 10x+6

Polynomial: [1,1,0,-1,-1] ➡ -x⁴-x³+x+1
Derivative: [1,0,-3,-4] ➡ -4x³-3x²+1

Sidef

<lang ruby>func derivative(f) {

   Poly(f.coeffs.map_2d{|e,k| [e-1, k*e] }.flat...)

}

var coeffs = [

   [5],
   [4,-3],
   [-1,6,5],
   [-4,3,-2,1],
   [-1, 6, 5],
   [1,1,0,-1,-1],

]

for c in (coeffs) {

   var poly = Poly(c.flip)
   var derv = derivative(poly)

   var d = { derv.coeff(_) }.map(0..derv.degree)

   say "Polynomial : #{'%20s' % c} = #{poly}"
   say "Derivative : #{'%20s' % d} = #{derv || 0}\n"

}</lang>

Output:

Polynomial :                  [5] = 5
Derivative :                  [0] = 0

Polynomial :              [4, -3] = -3*x + 4
Derivative :                 [-3] = -3

Polynomial :           [-1, 6, 5] = 5*x^2 + 6*x - 1
Derivative :              [6, 10] = 10*x + 6

Polynomial :       [-4, 3, -2, 1] = x^3 - 2*x^2 + 3*x - 4
Derivative :           [3, -4, 3] = 3*x^2 - 4*x + 3

Polynomial :           [-1, 6, 5] = 5*x^2 + 6*x - 1
Derivative :              [6, 10] = 10*x + 6

Polynomial :    [1, 1, 0, -1, -1] = -x^4 - x^3 + x + 1
Derivative :       [1, 0, -3, -4] = -4*x^3 - 3*x^2 + 1

Wren

<lang ecmascript>var derivative = Fn.new { |p|

   if (p.count == 1) return [0]
   var d = p[1..-1].toList
   for (i in 0...d.count) d[i] = p[i+1] * (i + 1)
   return d

}

var ss = ["", "", "\u00b2", "\u00b3", "\u2074", "\u2075", "\u2076", "\u2077", "\u2078", "\u2079"]

// for n <= 20 var superscript = Fn.new { |n| (n < 10) ? ss[n] : (n < 20) ? ss[1] + ss[n - 10] : ss[2] + ss[0] }

var polyPrint = Fn.new { |p|

   if (p.count == 1) return p[0].toString
   var terms = []
   for (i in 0...p.count) {
       if (p[i] == 0) continue
       var c = p[i].toString
       if (i > 0 && p[i].abs == 1) c = (p[i] == 1) ? "" : "-"
       var x = (i > 0) ? "x" : ""
       terms.add("%(c)%(x)%(superscript.call(i))")
   }
   return terms[-1..0].join("+").replace("+-", "-")

}

System.print("The derivatives of the following polynomials are:\n") var polys = [ [5], [4, -3], [-1, 6, 5], [-4, 3, -2, 1], [1, 1, 0, -1, -1] ] for (poly in polys) {

   var deriv = derivative.call(poly)
   System.print("%(poly) -> %(deriv)")

} System.print("\nOr in normal mathematical notation:\n") for (poly in polys) {

   var deriv = derivative.call(poly)
   System.print("Polynomial : %(polyPrint.call(poly))")
   System.print("Derivative : %(polyPrint.call(deriv))\n")

}</lang>

Output:

The derivatives of the following polynomials are:

[5] -> [0]
[4, -3] -> [-3]
[-1, 6, 5] -> [6, 10]
[-4, 3, -2, 1] -> [3, -4, 3]
[1, 1, 0, -1, -1] -> [1, 0, -3, -4]

Or in normal mathematical notation:

Polynomial : 5
Derivative : 0

Polynomial : -3x+4
Derivative : -3

Polynomial : 5x²+6x-1
Derivative : 10x+6

Polynomial : x³-2x²+3x-4
Derivative : 3x²-4x+3

Polynomial : -x⁴-x³+x+1
Derivative : -4x³-3x²+1

XPL0

<lang XPL0>int IntSize, Cases, Case, Len, Deg, Coef; [IntSize:= @Case - @Cases; Cases:=[[ 5],

       [ 4, -3],
       [-1,  6,  5],
       [-4,  3, -2,  1],
       [ 1,  1,  0, -1, -1],
       [ 0]];

for Case:= 0 to 5-1 do

   [Len:= (Cases(Case+1) - Cases(Case)) / IntSize;
   for Deg:= 0 to Len-1 do
       [Coef:= Cases(Case, Deg);
       if Deg = 0 then Text(0, "[")
       else    [IntOut(0, Coef*Deg);
               if Deg < Len-1 then
                   Text(0, ", ");
               ];
       ];
   Text(0, "]^M^J");
   ];

]</lang>

Output:

[]
[-3]
[6, 10]
[3, -4, 3]
[1, 0, -3, -4]