Calculus with polynomials

From Exampleproblems

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

$\frac{d}{dx} \sum^n_{k=0} a_k x^k = \sum^n_{k=0} ka_kx^{k-1}$

$\int \sum^n_{k=0} a_k x^k\,dx= \sum^n_{k=0} \frac{a_k x^{k+1}}{k+1} + c.$

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

1 Proof
2 Remark
3 Formal differentiation
4 References

Proof

The power rule for differentiation is:

For every natural number n, the derivative of

f (x) = x n

f'(x) = n x n - 1

To prove this rule, we use the definition of the derivative as a limit:

$f'(x) = \lim_{h\rarr0} \frac{f(x+h)-f(x)}{h}.$

Substituting $f (x) = x n$ gives

$f'(x) = \lim_{h\rarr0} \frac{(x+h)^n-x^n}{h}.$

Now, if you get the $x + h$ out of the brackets using the binomial formula, it would lead to

$f'(x) = \lim_{h\rarr0} \frac{x^n+nx^{n-1}h+\frac{1}{2}n(n-1)x^{n-2}h^2+\cdots-x^n}{h}.$

Now then, if we look carefully, we see that xⁿ and −xⁿ cancel eachother out, so this becomes:

$f'(x) = \lim_{h\rarr0} \frac{nx^{n-1}h+\frac{1}{2}n(n-1)x^{n-2}h^2+\cdots}{h}.$

We can even work out more things out of this big sum. We see that h appears in a lot of places, so let's get some out!

$f'(x) = \lim_{h\rarr0} \, \Big(nx^{n-1}+\frac{1}{2}n(n-1)x^{n-2}h+\cdots\Big).$

Right, now we just make h (the difference in x) go to zero. All terms but the first one disappear, and we get the result that we want:

$f'(x) = nx^{n-1}. \,$

Finally, to differentiate arbitrary polynomials, we use that differentiation is linear, so we have:

$\frac{d\left( \sum_{r=0}^n a_r x^r \right)}{dx} = \sum_{r=0}^n \frac{d\left(a_r x^r\right)}{dx} = \sum_{r=0}^n a_r \frac{d\left(x^r\right)}{dx} = \sum_{r=0}^n ra_rx^{r-1}.$

Remark

One can prove that the power rule is valid for any real exponent, that is

$\frac{d}{dx} \left(x^a\right) = ax^{a-1}$

for any real number a as long as x is in the domain of the functions on the left and right hand sides. Using this formula one can differentiate linear combinations of powers of x which are not necessarily polynomials.

Formal differentiation

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

Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). Calculus of a Single Variable: Early Transcendental Functions (3rd edition). Houghton Mifflin Company. ISBN 061822307X.ko:다항식의 미적분

th:แคลคูลัสกับพหุนาม

Retrieved from "http://www.exampleproblems.com/wiki/index.php/Calculus_with_polynomials"

Category: Calculus