Calculus with polynomials

Helping orphans the way you would do it

In mathematics, polynomials are perhaps the simplest functions with which to do calculus. Their derivatives and integrals are given by the following rules:

Hence the derivative of x¹⁰⁰ is 100x⁹⁹ and the integral of x¹⁰⁰ is x¹⁰¹/101 + c.

Table of contents

1 Proof
2 Generalisations
3 References
4 See also

Proof

Because differentiation is linear, we have:

So it remains to find for any natural number r. The derivative of function f(x) is given by Newton's difference quotient

By the binomial theorem,

and therefore

The derivative is the limit of this as

which gives the claimed result.

Generalisations

is generally true for all values of k where x^k is meaningful. In particular it holds for all rational k for values of x where x^k is defined.

If one has polynomials with coefficients that are not real or complex numbers (perhaps they are integers, or numbers modulo a prime number) then one can formally define the derivative according to the rules given above. This is useful, for example, in determining whether a polynomial will have multiple roots: compute the greatest common divisor of the polynomial and its formal derivative. If this polynomial is zero, then the original polynomial cannot have any multiple roots.

References

Calculus of a Single Variable: Early Transcendental Functions (3rd Edition) by Edwards, Hostetler, and Larson (2003) ISBN 0618226877