# polyder

Polynomial differentiation

## Syntax

``k = polyder(p)``
``k = polyder(a,b)``
``````[q,d] = polyder(a,b)``````

## Description

example

````k = polyder(p)` returns the derivative of the polynomial represented by the coefficients in `p`,$k\left(x\right)=\frac{d}{dx}p\left(x\right)\text{\hspace{0.17em}}.$```

example

````k = polyder(a,b)` returns the derivative of the product of the polynomials `a` and `b`,$k\left(x\right)=\frac{d}{dx}\left[a\left(x\right)b\left(x\right)\right]\text{\hspace{0.17em}}.$```

example

``````[q,d] = polyder(a,b)``` returns the derivative of the quotient of the polynomials `a` and `b`,$\frac{q\left(x\right)}{d\left(x\right)}=\frac{d}{dx}\left[\frac{a\left(x\right)}{b\left(x\right)}\right]\text{\hspace{0.17em}}.$```

## Examples

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Create a vector to represent the polynomial $p\left(x\right)=3{x}^{5}-2{x}^{3}+x+5$.

`p = [3 0 -2 0 1 5];`

Use `polyder` to differentiate the polynomial. The result is $q\left(x\right)=15{x}^{4}-6{x}^{2}+1$.

`q = polyder(p)`
```q = 1×5 15 0 -6 0 1 ```

Create two vectors to represent the polynomials $a\left(x\right)={x}^{4}-2{x}^{3}+11$ and $b\left(x\right)={x}^{2}-10x+15$.

```a = [1 -2 0 0 11]; b = [1 -10 15];```

Use `polyder` to calculate

`$q\left(x\right)=\frac{d}{dx}\left[a\left(x\right)b\left(x\right)\right].$`

`q = polyder(a,b)`
```q = 1×6 6 -60 140 -90 22 -110 ```

The result is

`$q\left(x\right)=6{x}^{5}-60{x}^{4}+140{x}^{3}-90{x}^{2}+22x-110.$`

Create two vectors to represent the polynomials in the quotient,

`$\frac{{x}^{4}-3{x}^{2}-1}{x+4}.$`

```p = [1 0 -3 0 -1]; v = [1 4];```

Use `polyder` with two output arguments to calculate

`$\frac{q\left(x\right)}{d\left(x\right)}=\frac{d}{dx}\left[\frac{p\left(x\right)}{v\left(x\right)}\right].$`

`[q,d] = polyder(p,v)`
```q = 1×5 3 16 -3 -24 1 ```
```d = 1×3 1 8 16 ```

The result is

`$\frac{q\left(x\right)}{d\left(x\right)}=\frac{3{x}^{4}+16{x}^{3}-3{x}^{2}-24x+1}{{x}^{2}+8x+16}.$`

## Input Arguments

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Polynomial coefficients, specified as a vector. For example, the vector `[1 0 1]` represents the polynomial ${x}^{2}+1$, and the vector `[3.13 -2.21 5.99]` represents the polynomial $3.13{x}^{2}-2.21x+5.99$.

Data Types: `single` | `double`
Complex Number Support: Yes

Polynomial coefficients, specified as two separate arguments of row vectors.

Example: `polyder([1 0 -1],[10 2])`

Data Types: `single` | `double`
Complex Number Support: Yes

## Output Arguments

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Differentiated polynomial coefficients, returned as a row vector.

Numerator polynomial, returned as a row vector.

Denominator polynomial, returned as a row vector.