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 = 1×3

     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 = 1×5

     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 $$b(x)=2x^2+4x+6$$.

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

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

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

q = 1×3

    -2    -8    -2

d = 1×5

     4    16    40    48    36