Calculus with polynomials

Topics in calculus

Function | Limits of functions | Continuity | Calculus with polynomials

Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem

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

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 using the C-notation of combinations,

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