In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, the power rule is used to differentiate functions of the form
, whenever
is a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
. Since
differentiation is a
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
operation on the space of differentiable functions,
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s can also be differentiated using this rule. The power rule underlies the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
as it relates a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
with a function's
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s.
Statement of the power rule
Let
be a function satisfying
for all
, where
. Then,
:
The power rule for integration states that
:
for any real number
. It can be derived by inverting the power rule for differentiation. In this equation C is
any constant.
Proofs
Proof for real exponents
To start, we should choose a working definition of the value of
, where
is any real number. Although it is feasible to define the value as the limit of a sequence of rational powers that approach the irrational power whenever we encounter such a power, or as the least upper bound of a set of rational powers less than the given power, this type of definition is not amenable to differentiation. It is therefore preferable to use a functional definition, which is usually taken to be
for all values of
, where
is the
natural exponential function and
is
Euler's number. First, we may demonstrate that the derivative of
is
.
If
, then
, where
is the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
function, the inverse function of the exponential function, as demonstrated by Euler. Since the latter two functions are equal for all values of
, their derivatives are also equal, whenever either derivative exists, so we have, by the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
,
or
, as was required.
Therefore, applying the chain rule to
, we see that
which simplifies to
.
When
, we may use the same definition with
, where we now have
. This necessarily leads to the same result. Note that because
does not have a conventional definition when
is not a rational number, irrational power functions are not well defined for negative bases. In addition, as rational powers of −1 with even denominators (in lowest terms) are not real numbers, these expressions are only real valued for rational powers with odd denominators (in lowest terms).
Finally, whenever the function is differentiable at
, the defining limit for the derivative is:
which yields 0 only when
is a rational number with odd denominator (in lowest terms) and
, and 1 when r = 1. For all other values of r, the expression
is not well-defined for
, as was covered above, or is not a real number, so the limit does not exist as a real-valued derivative. For the two cases that do exist, the values agree with the value of the existing power rule at 0, so no exception need be made.
The exclusion of
the expression (the case x = 0) from our scheme of exponentiation is due to the fact that the function
has no limit at (0,0), since
approaches 1 as x approaches 0, while
approaches 0 as y approaches 0. Thus, it would be problematic to ascribe any particular value to it, as the value would contradict one of the two cases, dependent on the application. It is traditionally left undefined.
Proofs for integer exponents
Proof by induction (natural numbers)
Let
. It is required to prove that
The base case may be when
or
, depending on how the set of
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
is defined.
When
,
When
,
Therefore, the base case holds either way.
Suppose the statement holds for some natural number ''k'', i.e.
When
,
By the principle of mathematical induction, the statement is true for all natural numbers ''n''.
Proof by binomial theorem (natural numbers)
Let
, where
.
Then,
Generalization to negative integer exponents
For a negative integer ''n'', let
so that ''m'' is a positive integer.
Using the
reciprocal rule,
In conclusion, for any integer
,
Generalization to rational exponents
Upon proving that the power rule holds for integer exponents, the rule can be extended to rational exponents.
Proof by
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
This proof is composed of two steps that involve the use of the chain rule for differentiation.
# Let
, where
. Then
. By the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
,
. Solving for
,
Thus, the power rule applies for rational exponents of the form
, where
is a nonzero natural number. This can be generalized to rational exponents of the form
by applying the power rule for integer exponents using the chain rule, as shown in the next step.
# Let
, where
so that
. By the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
,
From the above results, we can conclude that when
is a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
,
Proof by implicit differentiation
A more straightforward generalization of the power rule to rational exponents makes use of implicit differentiation.
Let
, where
so that
.
Then,
Differentiating both sides of the equation with respect to
,
Solving for
,
Since
,
Applying laws of exponents,
Thus, letting
, we can conclude that
when
is a rational number.
History
The power rule for integrals was first demonstrated in a geometric form by Italian mathematician
Bonaventura Cavalieri in the early 17th century for all positive integer values of
, and during the mid 17th century for all rational powers by the mathematicians
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
,
Evangelista Torricelli
Evangelista Torricelli ( , also , ; 15 October 160825 October 1647) was an Italian physicist and mathematician, and a student of Galileo. He is best known for his invention of the barometer, but is also known for his advances in optics and wo ...
,
Gilles de Roberval,
John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
, and
Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer.
He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earliest ...
, each working independently. At the time, they were treatises on determining the area between the graph of a rational power function and the horizontal axis. With hindsight, however, it is considered the first general theorem of calculus to be discovered.
The power rule for differentiation was derived by
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
and
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
, each independently, for rational power functions in the mid 17th century, who both then used it to derive the power rule for integrals as the inverse operation. This mirrors the conventional way the related theorems are presented in modern basic calculus textbooks, where differentiation rules usually precede integration rules.
Although both men stated that their rules, demonstrated only for rational quantities, worked for all real powers, neither sought a proof of such, as at the time the applications of the theory were not concerned with such exotic power functions, and questions of convergence of infinite series were still ambiguous.
The unique case of
was resolved by Flemish Jesuit and mathematician
Grégoire de Saint-Vincent and his student
Alphonse Antonio de Sarasa
Alphonse Antonio de Sarasa was a Jesuit mathematician who contributed to the understanding of logarithms, particularly as areas under a hyperbola.
Alphonse de Sarasa was born in 1618, in Nieuwpoort in Flanders. In 1632 he was admitted as a no ...
in the mid 17th century, who demonstrated that the associated definite integral,
:
representing the area between the rectangular hyperbola
and the x-axis, was a logarithmic function, whose base was eventually discovered to be the transcendental number
e. The modern notation for the value of this definite integral is
, the natural logarithm.
Generalizations
Complex power functions
If we consider functions of the form
where
is any complex number and
is a complex number in a slit complex plane that excludes the
branch point
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, ...
of 0 and any branch cut connected to it, and we use the conventional multivalued definition
, then it is straightforward to show that, on each branch of the complex logarithm, the same argument used above yields a similar result:
.
In addition, if
is a positive integer, then there is no need for a branch cut: one may define
, or define positive integral complex powers through complex multiplication, and show that
for all complex
, from the definition of the derivative and the binomial theorem.
However, due to the multivalued nature of complex power functions for non-integer exponents, one must be careful to specify the branch of the complex logarithm being used. In addition, no matter which branch is used, if
is not a positive integer, then the function is not differentiable at 0.
See also
*
*
*
*
*
*
*
*
References
Notes
Citations
Further reading
* Larson, Ron; Hostetler, Robert P.; and Edwards, Bruce H. (2003). ''Calculus of a Single Variable: Early Transcendental Functions'' (3rd edition). Houghton Mifflin Company. .
{{Calculus topics
Articles containing proofs
Differentiation rules
Mathematical identities
Theorems in analysis
Theorems in calculus