# Integrate and Differentiate Polynomials

This example shows how to use the polyint and polyder functions to analytically integrate or differentiate any polynomial represented by a vector of coefficients.

Use polyder to obtain the derivative of the polynomial $p\left(x\right)={x}^{3}-2x-5$. The resulting polynomial is $q\left(x\right)=\frac{d}{dx}p\left(x\right)=3{x}^{2}-2$.

p = [1 0 -2 -5];
q = polyder(p)
q = 1×3

3     0    -2

Similarly, use polyint to integrate the polynomial $p\left(x\right)=4{x}^{3}-3{x}^{2}+1$. The resulting polynomial is $q\left(x\right)=\int p\left(x\right)dx={x}^{4}-{x}^{3}+x$.

p = [4 -3 0 1];
q = polyint(p)
q = 1×5

1    -1     0     1     0

polyder also computes the derivative of the product or quotient of two polynomials. For example, create two vectors to represent the polynomials $a\left(x\right)={x}^{2}+3x+5$ and $b\left(x\right)=2{x}^{2}+4x+6$.

a = [1 3 5];
b = [2 4 6];

Calculate the derivative $\frac{d}{dx}\left[a\left(x\right)b\left(x\right)\right]$ by calling polyder with a single output argument.

c = polyder(a,b)
c = 1×4

8    30    56    38

Calculate the derivative $\frac{d}{dx}\left[\frac{a\left(x\right)}{b\left(x\right)}\right]$ by calling polyder with two output arguments. The resulting polynomial is

$\frac{d}{dx}\left[\frac{a\left(x\right)}{b\left(x\right)}\right]=\frac{-2{x}^{2}-8x-2}{4{x}^{4}+16{x}^{3}+40{x}^{2}+48x+36}=\frac{q\left(x\right)}{d\left(x\right)}.$

[q,d] = polyder(a,b)
q = 1×3

-2    -8    -2

d = 1×5

4    16    40    48    36

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