Product rule

The product rule, or Leibniz rule ( by GW Leibniz ) is a fundamental rule of differential calculus. It returns the calculation of the derivative of a product of functions to calculating the derivatives of the individual functions.

An application of the product rule in calculus is the method of partial integration. In the event that one of the two functions is constant, the product typically moves on to the simpler factor rule.

Statement of the product rule

Are the functions and of an interval in the set of real or complex numbers in one place differentiable, so also is by

Defined function differentiable at the point, and it is

Or in short:

Application Examples

In the following we always

• Is and is obtained from the knowledge of the product rule and the statement

• If so and is therefore is

Statement and proof

The product of two real ( in a location of differentiable ) functions, and has at the location of the value which can be interpreted as the surface area of a rectangle with the sides. If going all the changes around and around The change of the surface area is then ( see figure ) is composed of

Dividing by the following stress

The difference quotient of the product or area function at the point

For against also strives (and thus the whole last term ) against so you at the point

Receives, as claimed. This is essentially the argument as it is found in a first proof of the product rule in 1677 in a manuscript of Leibniz. The product rule, which he proves there together with the quotient rule, so that was one of the first rules for the application of the infinitesimal calculus, which he derived. However, he did not use a limit, but still differentials and concluded that disappears because it is infinitesimally small compared to the other summands. Euler still using the same argument, only when there is a Cauchy evidence with threshold values:

Consider the function by the derivative of at a point is then given by the limit of the difference quotient

Given. Delivers addition and subtraction of the term

The execution of the two border crossings delivering the product rule

Generalizations

Products of vectors and matrix-vector product

In the proof of the product rule ( sums, differences, products with numbers ) are calculated from the values ​​of linear combinations formed, nor from the values ​​of the roles of and are clearly separated: is the left factor of the right. The evidence therefore carries over to all product defects, which are linear in both the left and the right factor. In particular, the product rule also applies to

• Scalar 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 be successively applied to several factors. so would

In general, for a function which can be written as a product of functions, the derivative

Have the functions no zeros, so you can this rule also in the clear form

Write; Such fractures are called logarithmic derivatives.

Higher Derivatives

The rule for derivatives of order a product of two functions Leibniz was already known and is sometimes also referred to as corresponding Leibniz rule. It results from the product rule by induction to

The expressions of the form occurring here are the binomial coefficients. The above formula contains the actual product rule as a special case. She has striking similarity to the binomial theorem

This similarity is no coincidence, the usual induction proof is completely analogous in both cases; one can prove the Leibniz rule, but also by means of the binomial theorem.

Higher-dimensional domain

Is generalized to functions with höherdimensionalem definition range, this is how the product rule formulated as follows: Let be an open subset, differentiable functions and a direction vector. Then the product rule applies to the directional derivative:

According applies to the gradient

In the language of differentiable manifolds in question, these two statements:

• If a tangent and locally differentiable functions, then applies

• Are locally differentiable functions, then the following relation between the outer derivations applies:

Higher partial derivatives

Be Then:

Holomorphic functions

The product rule is also valid for complex differentiable functions: It is and holomorphic. Then is holomorphic, and it is

General differentiable maps

Let be an open interval, is a Banach algebra (eg the algebra of real or complex matrices ) and differentiable functions. Then we have

Thereby designated " · " the multiplication in the Banach algebra.

Are general and Banach spaces, and differentiable functions, so also applies a product rule, where the function of the product is taken over by a bilinear form. From this it is required that it is continuous, so limited:

With a fixed constant. Then the product rule

Corresponding statements hold for higher-dimensional domains.

Abstraction: derivations

General called pictures which the product rule

Meet, derivative ions. ( The order of the factors is here the case of a Derivation chosen with an algebra and a left - module. )

In connection with - or -graded algebras ( " Superalgebren " ), however, the concept of derivation must be replaced by the Antiderivation; the corresponding equation is then

For homogeneous elements it refers to the degree of the most prominent example of a Antiderivation is the exterior derivative of differential forms