Calculus with polynomials

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 Generalisation
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.

Generalisation

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.

References

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