NationMaster - Encyclopedia: Calculus with polynomials

In mathematics, polynomials are perhaps the simplest functions with which to do calculus. Their derivatives and indefinite integrals are given by the following rules: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... In mathematics, polynomial functions, or polynomials, aresimple and smooth [[function_(mathematics) | functions] an important class of ]. Here, simple means they are constructed using only multiplication and addition. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique type of another (possibly the same) set (the codomain, not to be confused with the range). ... Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... In mathematics, the derivative is one of the two central concepts of calculus. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

$left( sum^n_{k=0} a_k x^kright)' = sum^n_{k=0} ka_kx^{k-1}$

and

$int!left( sum^n_{k=0} a_k x^kright),dx= sum^n_{k=0} frac{a_k x^{k+1}}{k+1} + c.$

Hence, the derivative of $x 100$ is 100x⁹⁹ and the integral of $x 100$ is $x 101 / 101 + c .$

This article will state and prove the power rule for differentiation, and then use it to prove these two formulas.

1 The power rule
2 Proof of the power rule
3 Differentiation of arbitrary polynomials
4 Generalization
5 See also
6 References

The power rule

The power rule for differentiation states that for every natural number n, the derivative of $f (x) = x n$ is $f'(x) = n x n - 1,$ that is, Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), and they can be used for ordering (this is...

$left(x^nright)'=nx^{n-1}.$

The power rule for integration

$int! x^n , dx=frac{x^{n+1}}{n+1}+C$

is then an easy consequence. One just needs to take the derivative of this equality and use the power rule and linearity of differentiation on the right-hand side. In mathematics, a linear transformation (also called linear operator <<wrong! operators are LTs on the same vector space or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

Proof of the power rule

To prove the power rule for differentiation, we use the definition of the derivative as a limit: In mathematics, the derivative is one of the two central concepts of calculus. ... In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...

$f'(x) = lim_{hrarr0} frac{f(x+h)-f(x)}{h}.$

Substituting $f (x) = x n$ gives

$f'(x) = lim_{hrarr0} frac{(x+h)^n-x^n}{h}.$

One can then express $(x + h) n$ by applying the binomial theorem to obtain In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

$f'(x) = lim_{hrarr0} frac{sum_{i=0}^{n} {{n choose i} x^i h^{n-i}}-x^n}{h}.$

The $i = n$ term of the sum can then be written independently of the sum to yield

$f'(x) = lim_{hrarr0} frac{sum_{i=0}^{n - 1} {{n choose i} x^i h^{n-i}} + x^n -x^n}{h}.$

Canceling the $x n$ terms one generates

$f'(x) = lim_{hrarr0} frac{sum_{i=0}^{n - 1} {{n choose i} x^i h^{n-i}}}{h}.$

An $h$ can be factored out from each term in the sum to give

$f'(x) = lim_{hrarr0} frac{hsum_{i=0}^{n - 1} {{n choose i} x^i h^{n-i-1}}}{h}.$

From thence we can cancel the $h$ in the denominator to obtain

$f'(x) = lim_{hrarr0} sum_{i=0}^{n - 1} {{n choose i} x^i h^{n-i-1}}.$

To evaluate this limit we observe that $n - i - 1 > 0$ for all $i < n - 1$ and equal to zero for $i = n - 1.$ Thus only the $h 0$ term will survive with $i = n - 1$ yielding

$f'(x) = {n choose {n-1}} x^{n-1}.$

Evaluating the binomial coefficient gives In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ...

${n choose {n-1}} = frac{n!}{(n-1)! 1!} = frac{n (n-1)!}{(n-1)!} = n.$

It follows that

$f'(x) = n x^{n-1}. !$

Differentiation of arbitrary polynomials

To differentiate arbitrary polynomials, one can use the linearity property of the differential operator to obtain: In mathematics, a linear transformation (also called linear operator <<wrong! operators are LTs on the same vector space or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

$left( sum_{r=0}^n a_r x^r right)' = sum_{r=0}^n left(a_r x^rright)' = sum_{r=0}^n a_r left(x^rright)' = sum_{r=0}^n ra_rx^{r-1}.$

Using the linearity of integration and the power rule for integration, one shows in the same way that

$int!left( sum^n_{k=0} a_k x^kright),dx= sum^n_{k=0} frac{a_k x^{k+1}}{k+1} + c.$

Generalization

One can prove that the power rule is valid for any real exponent, that is In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

$left(x^aright)' = ax^{a-1}$

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, together with

$int ! x^{-1}, dx= ln x+c,$

one can differentiate and integrate linear combinations of powers of x which are not necessarily polynomials.

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.

Category: Calculus

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Contents

The power rule GA_googleFillSlot("encyclopedia_square");

Proof of the power rule

Differentiation of arbitrary polynomials

Generalization

See also

References

The power rule