Product rule
The product rule or Leibniz rule (after G. W. Leibniz ) is a fundamental rule of differential calculus . It reduces the calculation of the derivative of a product of functions to the calculation of the derivatives of the individual functions.
One application of the product rule in integral calculus is the method of partial integration . In the event that one of the two functions is constant, the product rule changes over to the simpler factor rule .
Statement of the product rule
If the functions and are differentiable from an interval into the set of real or complex numbers at one point , then that is also through
- for all
defined function at the point differentiable, and it applies
or short:
Application examples
In the following always be
- Is and so one receives the statement from the knowledge of and with the product rule
- Is and so is so is
- and by reshaping you get the statement
If you use the short notation you get, for example, the derivation of the following function
When multiplied it results
Explanation and evidence
The product of two real functions (which can be differentiated at one point ) and has at that point the value that can be interpreted as the area of a rectangle with the sides and . If going to such changes in order and to the change of the area will come back (see figure) composed of
If you divide by then you get with
the difference quotient of the product or area function at the point
For against also strives (and thus the whole last summand) against so that one is at the point
received as claimed. This is also essentially the line of argument found in a first proof of the product rule in 1677 in a Leibniz manuscript. The product rule, which he proves there together with the quotient rule , was thus one of the first rules for the application of the infinitesimal calculus that he derived. However, he did not use a limit value , but also differentials and concluded that it is not applicable because it is infinitesimally small in comparison to the other summands. Euler used the same argument, only with Cauchy is there a proof with limit values:
The function is given by The derivation of at one point is then by the limit value of the difference quotient
given. Addition and subtraction of the term provides
Executing the two border crossings provides the product rule
Generalizations
Vector products and matrix vector products
When proving the product rule, linear combinations (sums, differences, products with numbers) are formed from the values, as are the values of the roles of and are clearly separated: is the left factor, the right one. The proof therefore carries over to all product formations that are linear in both the left and the right factor. In particular, the product rule also applies to
- Dot products of two vectors
- Vector products (cross products) of two vectors
- Matrix vector products .
Vectors or matrices are to be understood as functions of an independent variable.
More than two factors
The product rule can also be applied successively to several factors. That would be
- and
- etc.
In general , the derivative is for a function that can be written as a product of functions
If the functions have no zeros, this rule can also be used in a clear form
- (or short )
write; such fractions are called logarithmic derivatives .
Higher derivatives
The rule for derivatives of the -th order for a product of two functions was already known to Leibniz and is accordingly sometimes also referred to as Leibniz's rule . It results from the product rule by means of complete induction to
The expressions of the form appearing here are binomial coefficients . The above formula contains the actual product rule as a special case. It is strikingly similar to the binomial theorem
This similarity is no accident; the usual induction proof runs completely analogously in both cases; but one can also prove Leibniz's rule with the help of the binomial theorem.
For higher derivatives of more than two factors, the multinomial theorem can be transferred accordingly . The following applies:
Higher-dimensional domain of definition
If one generalizes to functions with a higher-dimensional domain, the product rule can be formulated as follows: Let there be an open subset , differentiable functions and a direction vector. Then the product rule for the directional derivative applies :
The same applies to the gradients
In the language of differentiable manifolds these two statements read:
- If a tangential vector and locally differentiable functions, then applies
- If functions are locally differentiable, then the following relationship applies between the outer derivatives :
Higher partial derivatives
Let then:
Holomorphic functions
The product rule also applies to complex differentiable functions: Let be and holomorphic . Then is holomorphic, and it holds
General differentiable maps
There are an open interval, a Banach (for example, the algebra of real or complex. - matrices ) and differentiable functions. Then:
"·" Denotes the multiplication in the Banach algebra.
If more general and Banach spaces , and differentiable functions, then a product rule also applies, whereby the function of the product is taken over by a bilinear form . This is required to be continuous , i.e. restricted :
- for all
with a fixed constant . Then the product rule applies
Corresponding statements apply to higher-dimensional domains of definition.
Leibniz rule for divided differences
The Leibniz rule can be applied to divided differences :
The special case
includes the original Leibniz rule.
Abstraction: derivations
In general, one calls images which the product rule
meet, derivatives. The order of the factors here is the case of a Derivation with algebra and one - Links module selected.
In connection with - or - graduated algebras (“ super algebras ”), however, the term derivation must be replaced by that of anti- derivation . The corresponding equation then reads
for homogeneous elements here denotes the degree of The most prominent example of an anti-derivation is the outer derivation for differential forms
literature
The product rule for functions is explained in any book that deals with differential calculus in a general way.
- Otto Forster : Analysis 1. Differential and integral calculus of a variable. 7th edition. Vieweg, Braunschweig 2004, ISBN 3-528-67224-2 .
- Otto Forster: Analysis 2. Differential calculus in R ^{n} . Ordinary differential equations. 6th edition. Vieweg, Braunschweig 2005, ISBN 3-528-47231-6 .
- Konrad Königsberger : Analysis. 2 vols. Springer, Berlin 2004, ISBN 3-540-41282-4 .
- CH Edwards Jr .: The Historical Development of the Calculus. Springer, New York 1979.
Web links
- Derivation of the product rule and numerous examples ( Memento from February 29, 2008 in the Internet Archive )
Individual evidence
- ↑ Lawrence C. Evans: Partial Differential Equations. ISBN 0-8218-0772-2 , 19th edition, p. 12.
- ^ De Boor: Divided Differences. Surveys in Approximation Theory. Volume 1, 2005, pp. 46-69.