Chain rule
The chain rule is one of the basic rules of differential calculus. She tells about the a function that itself can be represented as a concatenation of two differentiable functions dissipation. Core statement of the chain rule is that such a function is differentiable itself again and gives its derivative, by deriving the two interlinked functions separately - evaluated in the right places - multiplied together.
The chain rule can be generalized to functions that can be represented as a concatenation of more than two differentiable functions. Also, such a function is again differentiable, its derivative is obtained by multiplying the derivatives of all nested functions.
The chain rule is a special case of the multidimensional chain rule for the one-dimensional case.
Its counterpart in the integral calculus is the integration by substitution.
Mathematical formulation
Let U, V open intervals, and functions.
The function is differentiable at the point and is differentiable at the point.
Then the composite function
Differentiable at the point and we have:
In connection with the chain rule is also called the outer, the inner function of.
Practical rule of thumb: The derivative of a function formed by concatenation in point is the " exterior derivative " - evaluated at - times the derivative of the inner function - evaluated at the point. Or in short: " exterior derivative times inner derivation ".
Example
It is considered by the defined function.
This can be represented as a concatenation of the function
With the feature
Because it is true. In the terminology of the chain rule called the outer, the inner function. It is common to identify the sake of clarity in the external function, the independent variable of the function icon of the inner function, though the designation of the variable in principle irrelevant. You could equally write for the first function also.
For the application of the chain rule, we need the derivatives ( " exterior derivative ") and ( "inner derivation " ):
And
Since both are also differentiable, is also differentiable by the chain rule, and it applies to its derivative:
Well, so we get a total of:
Note that the representation of a function as a concatenation of an outer no means must be unique with an inner function. Thus, for the example function as a concatenation of the functions and interpret, because also applies to these two functions:
The application of the chain rule is computationally expensive in this case, since at least the term must be multiplied out.
Overall, it can be discovered even in this example, the chain rule in terms of the constructivist didactics. Multiplying out gives:
After deriving the inner function is dissected by factoring out:
From this it is then the chain rule suggests, the then yet to be proven in its generality.
Geometric illustration
From x to the function value u (v ( x)) can pass you by first calculating v (x) and u (v). The function v (x ), the slope of v '(x) (inner lead). U ( v) is the slope of u '( v) ( outer lead). The gradient of u ( V (x)) u '(x) (total extraction).
The term arises here with by extension of the fracture, ie multiplication with and transfer. It must be noted: the concatenation of functions is quite different than the multiplication of functions.
For the difference quotients (see figure).
Through the border crossing? X → 0, the differential quotient of the difference quotient. From the above figure shows: If Ax approaches zero, then DELTA.v.
One then obtains for the total of the concatenated function derivative:
Note: The notation used here with differentials (for example ) to Leibniz notation is equivalent to the above Lagrangian, see also the last paragraph of this article.
Evidence
Be
Because is differentiable applies
That is, at constant. Additionally, for all
It follows
Complex Functions
Be with open subsets, eg, areas and functions.
The function is differentiable at the point and is differentiable at the point.
Then the composite function
Differentiable at the point and we have:
Conclusion: The complex chain rule ( including its proof) completely analogous to the real case.
Generalization to multiple concatenations
Somewhat more complicated is the differentiation, when more than two functions are linked. In this case, the chain rule is applied recursively. For example, results from concatenating three functions u, v and w
The derivation
Generally has the function of
The derivation
As can be proved by induction. So in practical calculating the derivative multiplying factors arising as follows:
The first factor is obtained by expressing that the outermost function with one independent variable and derived. Instead of these independent variables, the arithmetic expression for the remaining (inner) is to use features. The second factor is calculated according to the derivation of the second outermost function, wherein the calculation expression is to be used for its internal functions, too. This procedure assumes you continue up to the last factor, the innermost derivation.
As an example may serve the turn function. This can be represented as a concatenation of the three functions:
Because it is:
This provides the generalized to multiple concatenations chain rule with
The derivation
Generalization for higher derivatives
A generalization for higher derivatives is much more complicated and difficult to prove. She is known as a formula of Faa di Bruno.
Generalization to functions of several variables and illustrations
Here we consider differentiable functions ( images ). The derivative at the point of such an image then a linear map, which can be represented by a matrix, the Jacobian matrix.
The chain rule states that the concatenation of two differentiable maps is differentiable again. Its derivative is obtained by concatenating the individual leads. The associated Jacobian matrix is the matrix product of the individual Jacobian matrices.
In detail: Are the figures in point and the point differentiable, then the concatenation in point is differentiable, and it is
And
In a similar form can be a chain rule for Fréchet derivatives of mappings between Banach spaces and for the derivatives ( differentials, Tangentialabbildungen ) formulate of maps between differentiable manifolds.
Different notations in physics and other sciences
In many natural sciences such as physics, and in engineering science, the chain rule is widely used. However, here is a special notation has evolved which deviates significantly from the mathematical notation the chain rule.
Presentation of the notation
In engineering literature for the derivative of a function on the variables in the rule, the notation
Preferred. Is a concatenation of two functions: with, this is how the chain rule in this notation:
It is also common convention to identify the independent variables of the function to the function of the inner function icon, but omitting all argument brackets:
Ultimately, no new icon for chaining introduced, but also identifies the entire chain with the outer function.
The chain rule then takes on the following appearance:
Formally, the chain rule here as an extension of " break " with is so that it in physical literature (and also in other natural sciences and engineering ) is common, not to mention the chain rule for application name. Instead, one often finds replacement formulations, it is about the " expansion of with " the speech, partly missing a justification completely. Even though this is not always at first sight is invisible to the untrained eye, behind all these formulations invariably the chain rule of differential calculus.
Although the presented notation breaks with some mathematical conventions, it enjoys great popularity, and widespread popularity because it allows with derivatives (at least loosely ) as with "normal breaks " can be expected. Many bills they designed also transparent, as brackets omitted and only very few symbols must be used. In many cases also the size described by a concatenation of a certain physical variable is (for example, a power or a voltage ) for which a particular letter "reserved" (about energy E and U is voltage). The above notation makes it possible to use this letter in the total bill throughout.
Example
The kinetic energy of a body depends on its velocity from V. The speed again depends on the time, so the kinetic energy of the body is a function of time determined by the concatenation of
Will be described. Would we calculate the change in kinetic energy after time, then by the chain rule
In physical literature one would find the last equation in the following ( or similar ) shape:
A clear advantage is the consistent use of function symbols whose letters match those of the underlying physically relevant quantity (E for energy, v for velocity).