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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 $$p(x)={x}^{3}-2x-5$$. The resulting polynomial is $$q(x)=\frac{d}{dx}p(x)=3{x}^{2}-2$$.

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

`q = `*1×3*
3 0 -2

Similarly, use `polyint`

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

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 $$a(x)={x}^{2}+3x+5$$ and $$b(x)=2{x}^{2}+4x+6$$.

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

Calculate the derivative $$\frac{d}{dx}[a(x)b(x)]$$ by calling `polyder`

with a single output argument.

c = polyder(a,b)

`c = `*1×4*
8 30 56 38

Calculate the derivative $$\frac{d}{dx}\left[\frac{a(x)}{b(x)}\right]$$ by calling `polyder`

with two output arguments. The resulting polynomial is

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

[q,d] = polyder(a,b)

`q = `*1×3*
-2 -8 -2

`d = `*1×5*
4 16 40 48 36

`conv`

| `deconv`

| `polyder`

| `polyint`