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

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

     3     0    -2

Similarly, use polyint to integrate the polynomial $p(x) = 4x^3 - 3x^2 + 1$. 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    -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 $b(x) = 2x^2 + 4x + 6$.

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

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

c = polyder(a,b)
c =

     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

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

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

    -2    -8    -2


d =

     4    16    40    48    36

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