In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, polynomial

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s are perhaps the simplest functions

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 used in calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

. Their derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s and indefinite integrals are given by the following rules:

and

Hence, the derivative of is and the indefinite integral of is where C is an arbitrary constant of integration

Arbitrary constant of integration

In calculus, the indefinite integral of a given function is only defined up to an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives...

.

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

Power rule

The power rule is for polynomials and states that for every integer n, the derivative of is that is,

The power rule for integration
for natural n is then an easy consequence. One just needs to take the derivative of this equality and use the power rule and linearity

Linear transformation

In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

 of differentiation on the right-hand side.

Proof

To prove the power rule for differentiation, we use the definition of the derivative as a limit

Limit of a function

In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

. But first, note the factorization for :


Using this, we can see that


Since the division has been eliminated and we have a continuous function, we can freely substitute to find the limit:


The case of is trivial because , so .

The use of the quotient rule allows the extension of this rule for n as a negative integer, and the use of the laws of exponents and the chain rule allows this rule to be extended to all rational values of n. For an irrational n, a rational approximation is appropriate.

Differentiation of arbitrary polynomials

To differentiate arbitrary polynomials, one can use the linearity property

Linear transformation

In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

 of the differential operator

Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...

to obtain:


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

Generalization

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


for any value a as long as x is in the domain of the functions on the left and right hand sides and a+1 is nonzero. Using this formula, together with
one can differentiate and integrate linear combinations of powers of x which are not necessarily polynomials.