# 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

## ALGOL 68

```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```
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

```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
```
Output:
```{ }
{ -3 }
{ 6 10 }
{ 3 -4 3 }
{ 1 0 -3 -4 }
```

The implementation of `pdiff`:

```USING: kernel math.vectors sequences ;
IN: math.polynomials
: pdiff ( p -- p' ) dup length <iota> v* rest ;
```

## 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
```
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
```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 + ss[n-10]
}
return ss + ss
}

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)
}
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")
}
}
```
Output:
```The derivatives of the following polynomials are:

 -> 
[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
```

```deriv = zipWith (*) [1..] . tail

main = mapM_ (putStrLn . line) ps
where
line p = "\np  = " ++ show p ++ "\np' = " ++ show (deriv p)
ps = [,[4,-3],[-1,6,5],[-4,3,-2,1],[1,1,0,-1,-1]]
```
```main

p  = 
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

```{-# 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 = [,[4,-3],[-1,6,5],[-4,3,-2,1],[1,1,0,-1,-1]]
```
``` 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.

```# The input should be a non-empty array of integers representing a polynomial.
# The output likewise represents its derivative.
def derivative:
. as \$p
| if length == 1 then 
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 . | 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 polys: [ , [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" ) ;

Output:
```Example polynomials and their derivatives:

 -> 
[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

```using Polynomials

testcases = [
("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
```
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

```use strict;
use warnings;
use feature 'say';
use utf8;
binmode(STDOUT, ':utf8');

sub pp {
my(@p) = @_;
return 0 unless @p;
my @f = \$p;
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 (, [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";
}
```
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

```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 , [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 '';
}
```
Output:
```Polynomial:  ➡ 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

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

var coeffs = [
,
[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"
}
```
Output:
```Polynomial :                   = 5
Derivative :                   = 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

```var derivative = Fn.new { |p|
if (p.count == 1) return 
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 + ss[n - 10] : ss + ss }

var polyPrint = Fn.new { |p|
if (p.count == 1) return p.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" : ""
}
return terms[-1..0].join("+").replace("+-", "-")
}

System.print("The derivatives of the following polynomials are:\n")
var polys = [ , [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")
}
```
Output:
```The derivatives of the following polynomials are:

 -> 
[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

```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");
];
]```
Output:
```[]
[-3]
[6, 10]
[3, -4, 3]
[1, 0, -3, -4]
```