Differential calculus

The differential or differential calculus is an integral part of analysis and thus an area of mathematics. It is closely related to the integral calculus, with which it is grouped together under the name calculus. The central theme of the differential calculus is the calculation of local changes of functions.

To this end, relevant and at the same fundamental concept of differential calculus is the derivative of a function (also called differential quotient ), whose geometric correspondence is the tangent slope. The derivation is ( according to the idea of ​​Leibniz ), the proportionality factor between vanishingly small ( infinitesimal ) changes the input value and the resulting also infinitesimal changes of the function value. Is there such a proportionality factor, so you call the function differentiable. Equivalent to the derivative at a point is defined as the linear transformation, the local best approximates the change of the linear function in all figures. Accordingly, the discharge is also called the linearization of the function.

In many cases, the differential calculus is an indispensable tool for the formation of mathematical models, which are intended to represent reality as accurately as possible, and for their subsequent analysis. The correspondence of the derivative in the matter under investigation is often the instantaneous rate of change; in economics we also speak often of marginal rates (eg, marginal cost, marginal productivity of a factor of production, etc.).

This article also explains the mathematical concepts: difference quotient, derivative, differentiation, continuously differentiable, smooth, partial derivatives, total derivative, reducing the degree of a polynomial.

In geometric language, the derivation is a generalized gradient. The geometric concept of slope is originally defined only for linear functions, the function graph is a straight line. Deriving a desired function at a point defined as to the slope of the tangent at the point of the graph.

In arithmetic language indicates the derivative of a function for each, how large the linear portion of the change (the change order 1) if changes by an arbitrarily small amount. For the exact formulation of this fact, the term limit is used (or limes ).

In a classical physical application, the derivation of the local or distance-time function delivers to the time the instantaneous velocity of a particle. The derivative of the instantaneous velocity with respect to time gives the instantaneous acceleration.

• 3.1 Example for the calculation of an elementary derivation function
• 3.2 Example of a not everywhere differentiable function
• 3.3 Example of a not everywhere continuously differentiable function
• 3.4 Derivation Rules

• 8.1 Calculation of minima and maxima 8.1.1 Horizontal tangents
• 8.1.2 Necessary and sufficient condition in the example
• 8.1.3 Curve Sketching

History

The task of the differential calculus was known as the tangent problem since ancient times. An obvious approach was to determine the tangent to a curve by its secant over a finite ( finite here means greater than zero ) to approximate but arbitrarily small interval. The technical difficulty was overcome, to be expected with such an infinitesimally small interval width. The first beginnings of the differential calculus dates back to Pierre de Fermat. He developed a method to 1628 to determine extreme points of algebraic terms and calculate tangents to conic sections and other curves. His "method" was purely algebraic. Fermat considered no border crossings and certainly no derivatives. Nevertheless, lets his "method" interpreted with modern means of analysis and justify and has mathematicians such as Newton and Leibniz inspired proven. Some years later, René Descartes chose a different algebraic access, by laying out a circle to a curve. This intersects the curve in two closely spaced points; unless it touches the curve. This approach enabled him to determine the slope of the tangent for special curves.

End of the 17th century did Isaac Newton and Gottfried Wilhelm Leibniz independently, free of contradictions to develop working calculi ( history of discovery and priority dispute see History of calculus ). However, Newton tackled the problem from a different angle as Leibniz. While Newton was concerned the problem physically on the instantaneous velocity problem, tried Leibniz geometrically over the tangent problem. Her work has allowed the abstraction of purely geometric notion and are therefore considered as the beginning of analysis. They became known primarily through the book of the nobles Guillaume François Antoine, Marquis de L' Hospital, who took private lessons from Johann Bernoulli and his research on analysis published so. The currently known derivation rules are based primarily on the works of Leonhard Euler, who coined the concept of function. Newton and Leibniz worked with arbitrarily small positive numbers. This has already been criticized by contemporaries as illogical, for example, by Bishop Berkeley in the polemic The analyst; or, a discourse Addressed to at infidel mathematician. The differential calculus but was further developed in spite of the prevailing uncertainty; primarily because of their numerous applications in physics and in other fields of mathematics. Symptomatic of that time was published by the Prussian Academy of Sciences in 1784 sweepstakes:

" ... The higher geometry often used infinitely large and infinitely small quantities; However, the ancient scholars have carefully avoided the infinite, and some famous analysts of our time confess that the words infinite size are contradictory. Thus, the Academy requires that you explain how so many correct sentences arising out of a contradictory assumption and that one SPECIFIED safe and clear basic concept, which is expected to replace the infinite, without making the statement too difficult or too long ... "

It was not until the early 19th century did Augustin -Louis Cauchy, to give the differential calculus the now common logical rigor by departed from the infinitesimals and defined the derivative as a limit of Sekantensteigungen ( difference quotient ). The definition of the limit used today was finally formulated by Karl Weierstrass end of the 19th century.

Definition

Introduction

Starting point for the definition of the derivative is the slope of the tangent approximation by a Sekantensteigung (sometimes called tendons slope ). Wanted is the slope of a function at a point. One first calculates the slope of the secant to over a finite interval:

The Sekantensteigung is the quotient of two differences; it is therefore also called difference quotient. With the shorthand notation for the Sekantensteigung can be abbreviated to write.

Difference quotients are well known from daily life, such as average speed:

To a tangent slope ( in the above example application that is an instantaneous velocity) to calculate, you have the two points through which the secant is drawn further and further back to each other. Here, go down as well as zero. The quotient is in many cases at. At this border crossing, the following definition is based:

Differentiability and derivative at a point: Formal Definition and notation

A function that maps an open interval U in the real numbers is differentiable at the point when the threshold

Exists. This limit is called the derivative or by deriving the location and is called

(pronounced "f of x is zero line ", " df from x to dx at the point x is equal to x is zero ", " df by dx of x is zero " respectively "d by dx of f of x is zero ").

The terms and are called differentials, but have (at least up to this point of theory) only symbolic meaning and they are allowed to date in the modern analysis only in this notation the formal quoted differential quotient. In some applications ( chain rule, integration of some differential equations, integration by substitution ) one expects them almost like "normal" variables. The precise formal justification for it provides the theory of differential forms. A differential is also of the usual notation for the integrals.

The notation of derivatives as the quotient of two differentials was introduced by Leibniz. Newton used a point about the size to be derived, what remains common in physics for time derivatives up today (, pronounced " x point "). The notation with bar () goes back to Joseph -Louis Lagrange, who introduced it in his book Théorie des fonctions analytiques 1797.

Over time, the following equivalent definition is found, which has proven in the more general context of complex or multi-dimensional functions as powerful:

A function is called differentiable at a point if there exists a constant such that

The growth of the function, if you look just a little away, approximately a value can, therefore be approximated by very good, is called the linear function with why the linearization of at the site.

Another definition is: There is with a steady at the point function and a constant such that for all valid

The conditions at the site and that is continuous, mean just that the " remainder term " converges to zero.

In both cases, the constant is uniquely defined and it is. The advantage of this formulation is that proofs are easier to perform, since no quotient must be considered. This representation of the best linear approximation was applied consistently been of Weierstrass, Henri Cartan and Jean Dieudonné.

If we denote a function as differentiable, without referring to a specific location, then this means the differentiability at every point of the domain, ie the existence of a unique tangent line at each point of the graph.

Every differentiable function is continuous, but the converse is not true. At the beginning of the 19th century was convinced that a continuous function at most a few places could not be differentiable (like the absolute value function ). Bernard Bolzano constructed it as the first mathematicians actually a function that is continuous but nowhere differentiable everywhere, which has been in the professional world, however, not known; Karl Weierstrass was then in the 1860's also such a function (see Weierstrass function ), which this time among mathematicians made ​​waves. A well-known example of a multidimensional continuous, not differentiable function is that of Helge von Koch in 1904 presented the Koch curve.

Deriving as a function of

The derivative of the function at the point designated by local describes the behavior of the function in the neighborhood of the point under consideration now will not be the only place that is differentiable. One can therefore try each number in the domain of definition of the derivative at this point (ie ) to assign. In this way, a new function whose domain is the set of all points to which is differentiable. This function is called the derivative function or just the derivative of and they say " is differentiable on ". For example, the square feature is at the desired position to derive the square function is thus differentiated to the set of real numbers. The corresponding derivative function is given by

The derivative function is normally different from the original one, the only exception is the multiple of the exponential function.

Is the derivative continuous, then is called continuously differentiable. Based on the name for the body ( the space ) of continuous functions with definition set is abbreviated with the space of continuously differentiable functions.

Calculation of discharges

Calculating the derivative of a function is called differentiation; ie, differentiating this function.

To calculate the derivative of elementary functions (eg, ...), one adheres closely to the definition given above, explicitly calculated a difference quotient and then can go to zero. In elementary mathematics, this is referred to as "h - " method. The typical user math does this calculation only a few times after in his life. Later, he knows the derivatives of the most important elementary functions by heart beats derivatives not quite as familiar features in a table work (eg Bronstein - Semendjajew or our Table of derivation and primitive functions ) to and computes the derivative of composite functions using the derivation rules.

Example of the calculation of an elementary derivation function

Wanted is the derivative of. Then we calculate the difference quotient as

And gets in the limit of the derivative of the function

Example of an everywhere differentiable function

Is not differentiable at the point 0:

Applies namely for all and thus

Other hand, applies to all and hence

Since the left - and right-sided limits do not match, the limit does not exist. The function is therefore not differentiable at the point considered. The differentiability of the function at all other points, however, is still given.

There are at the point 0, however, the right -hand derivative

And the left derivative

Looking at the graph of, one comes to the realization that the concept of differentiability clearly means that its graph passes kinks.

A typical example nowhere differentiable continuous functions whose existence at first seems hard to imagine, almost all paths of the Brownian motion. This is used for example to model the charts of stock prices.

Example of a not everywhere continuously differentiable function

A function is called continuously differentiable if its derivative is continuous. Even if the function is everywhere differentiable, the derivative does not have to be continuous. For example, the function

At every point, including, differentiable. The derivative, which can be determined at the point 0 on the difference quotient,

But is not continuous at 0.

Derivation rules

Derivatives of composite functions, such as or, one leads by means of derivation rules to the differentiation of elementary functions back (see also: Table of derivation and primitive functions ).

The following rules can be traced back the derivation of composite functions on derivations simpler functions. Be, and ( in the domain ) differentiable real functions and real numbers, then:

The Fundamental Theorem of Calculus

The key performance of Leibniz was the realization that related integration and differentiation. These he put in the main theorem of differential and integral calculus, also called Fundamental Theorem of Calculus. He says:

Is an interval that is a continuous function, and an arbitrary point, then the function

Continuously differentiable, and its derivative is.

This so a guide is provided for integrating: Find a function whose derivative is the integrand. Then:

Mean value theorem of differential calculus

Another key set of differential calculus is the mean value theorem, which was proved by Cauchy.

It is a function, which is (partly) defined in the closed interval, and continuously. Also, the function is differentiable in the open interval. Under these conditions, there is at least one, so

Applies.

Multiple derivations

Is the derivative of a function differentiable in turn, can be defined as the second derivative of a derivative of the first. In the same way, then third, fourth etc. derivations can be defined. A function can therefore easily differentiable, twice differentiable etc..

The second derivative can be interpreted as the geometric curvature of a graph. She has many physical applications. For example, the first derivative of the local time after the instantaneous velocity, the second derivative is the acceleration. From physics the notation, ( Say: point) comes, for discharges of any function with respect to time.

When politicians speak about the " decline of the rise in unemployment ," they speak of the second derivative ( change in slope ) in order to qualify the statement of the first derivative ( increase in the number of unemployed ).

Multiple recordings can be specified in three different ways:

Or physical event (with a derivative with respect to time)

Obviously, multi- apostrophe notation is preferred for low, one or the other number notation at high derivatives. For the formal designation of any derivatives are also specifies that and.

Taylor series and smoothness

If a () - times continuously differentiable function in the interval, then for all, and from the appearance of the so-called Taylor 's formula:

With the -th Taylor polynomial at the development site

And the ( ) th remainder term

An infinitely differentiable function is called a smooth function. Since it has all the derivatives, the above mentioned Taylor 's formula can be extended to the Taylor series of development with point

However, it turns out that the existence of all derivatives not appear that can be represented by the Taylor series. In other words: Any analytic function is smooth but not vice versa, as the given item in the Taylor series example of a non-analytic smooth function displays.

Often found in mathematical considerations the term sufficiently smooth. By this is meant that the function is differentiable as many times as necessary to carry out the current train of thought.

Applications

Calculation of minima and maxima

One of the most important applications of differential calculus is the determination of extreme values, mostly for the optimization of processes. These are, inter alia, monotonous functions at the edge of the domain, but generally at the points where the derivative is zero. A function may have a maximum or minimum value without the derivative exists at this point, but only at least locally differentiable functions are considered in the following. As an example we take the polynomial function with the function term

The figure shows the variation of the graphs of and.

Horizontal tangents

Has a function at a point its greatest value, thus for all this interval, and is the point differentiable, then the derivative can only be equal to zero there. A corresponding statement holds if takes the smallest value.

Geometric interpretation of this theorem of Fermat is that the graph of the function in local extreme points, a parallel to the axis tangent, also called horizontal tangent has.

It is thus for differentiable functions a necessary condition for the existence of an extreme point that the derivative at the point in question has the value 0:

Conversely, the fact that the derivative at a point is null, is not a guide to an extreme point, it could also exist, for example, a saddle point. A list of different reasonable criteria, the fulfillment of which can be safely close to an extreme point, is found in Article extreme value. These usually use the second or even higher derivatives.

Necessary and sufficient condition in the example

In the example,

It follows that applies exactly for and. The function values ​​at these points are, and, that the curve at the points and has a horizontal tangent, and only therein.

Since the sequence

Alternately consists of small and large values ​​, a high and a low point must be within this range. By the theorem of Fermat curve has a horizontal tangent at these points, so there are only the points identified above in Question: So is a high point and a low point.

Curve Sketching

Using the derivations still further properties of the function can be analyzed as turning points, saddle point, convexity or the above already mentioned monotony. The conduct of these tests is the subject of curve sketching.

Differential equations

Another important application of the differential calculus is the mathematical modeling of physical processes. Growth, movement or forces all have to do with derivatives, their formulaic description must therefore include differentials. Typically, this leads to equations in which derivations appear an unknown function, precisely differential equations.

For example, links the Newtonian law of motion

The acceleration of a body with its mass and the force acting on it. The basic problem of mechanics is therefore zurückzuschließen from a given acceleration to the local function of a body. This task, a reversal of the two-fold differentiation, has the mathematical form of a second order differential equation. The mathematical difficulty of this problem stems from the fact that location, velocity and acceleration are vectors that do not generally show the same direction, and that the force may depend on the time and the place.

Since many models are multidimensional, which later declared partial derivatives are often very important to let those formulated partial differential equations in the formulation. Mathematically, this compact described by differential operators and analyzed.

An example of applied differential calculus

Add the Microeconomics for example, various types of production functions are analyzed in order to draw lessons for macroeconomic contexts. This is especially the typical behavior of a production function of interest: how responsive the dependent variable output ( quantity produced of a good), when the input ( factor of production, such as labor or capital ) to a ( infinitesimally ) small unit is increased?

A basic type of production function is approximately the neoclassical production function. It is characterized by the fact that the output increases with each additional input, but that the gains are decreasing. For example, consider the production function for an operation

Prevail. The first derivative of this function is obtained by applying the chain rule

Since the radical expression of the first derivative can only be positive, we see that the yield increases with each additional input. The second derivative obtained

It is negative for all inputs, thus falling growth rates. So you could say that as the input to the output with proportional increases. The relative change of the output with respect to a relative change of the input is here given by the elasticity.

Differential calculus as a calculus

In addition to determining the slope of the differential calculus functions by their calculus is an essential tool in the term transformation. Here is dissolved itself from any connection with the original meaning of the derivative as the slope. If you have got two terms as identical can be by differentiation from others ( looking ) win identities. An example may illustrate this:

From the telescopic sum:

Shall

Be as simple as possible won. This is achieved by differentiation using the quotient rule:

Alternatively, the identity follows by multiplying out and then triple telescoping, but this is not so easy to see through.

Complex differentiability

So far spoken only of real functions. For differentiable functions with complex arguments, the definition is simply related to the linearization. Here the condition is much more restrictive than in the real: For example, the absolute value function, for example, complex differentiable nowhere. Same time, each in an environment once complex differentiable function is automatically infinitely differentiable, so there are all higher derivatives.

Derivatives of multidimensional functions

All previous versions put a function in a variable ( ie with a real or complex number as an argument ) is based. Functions that map vectors to vectors or vectors of numbers, can also have a derivation. However, a tangent to the graph function in these cases is no longer uniquely determined, since there are many different directions. Here, therefore, an extension of the previous derivation of the term is necessary.

Partial derivatives

→ Main article: Partial derivative

We first consider a function that goes by. An example is the temperature function: Depending on the location, the temperature is measured in the room, in order to assess how effective the heating. If the thermometer will move in a certain direction, a change in temperature is observed. This corresponds to the so-called directional derivative. The directional derivatives in specific directions, namely the coordinate axes are called the partial derivatives.

Overall, based on calculated total partial derivatives of a function in variables:

The individual partial derivatives of a function can also be bundled as a gradient or write Nablavektor. Partial derivatives can be again differentiable and its partial derivatives can be arranged then in the so-called Hessian matrix. Analogous to the one dimensional case, the candidates for extrema point where the derivative is zero, so the gradient disappears. Also analogously determine the second derivative, ie the Hessian matrix, in some cases, exactly the present case. In contrast to the one-dimensional, however, the diversity of forms in this case is larger. Using a principal axis transformation of the given by a multi-dimensional Taylor expansion in the considered point quadratic form can be classified the different cases.

Implicit differentiation

→ Main article: Implicit Differentiation

If a function is given by an implicit equation, it follows from the multidimensional chain rule, which applies to functions of several variables

For the function derivative therefore results

Total differentiability

→ Related article: Total differentiability

A function, an open set, is called a point total differentiable (or even differentiable ), if there exists a linear mapping, such that

For the one-dimensional case, this definition is consistent with the above. The linear map is uniquely determined in existence, is thus in particular independent of the choice of equivalent standards. The tangent is so abstracted by the local linearization of the function. The matrix representation of the first derivative of is called the Jacobian matrix. Is a matrix. For we obtain the gradient described above.

The following relationship exists between the partial derivatives and the total derivative: Is there a point in the total derivative, so there also exist all partial derivatives. In this case, the partial derivatives are in accordance with the coefficients of the Jacobian matrix. Conversely, not even follows from the existence of the partial derivatives at a point not necessarily the total differentiability, yes the continuity. If the partial derivatives but also continuous in a neighborhood of, then the function also is totally differentiable.

Important phrases

• Set of black: The Differentiationsreihenfolge is in the calculation of partial derivatives of higher order irrelevant if all partial derivatives up to this order (including ) are continuous.
• Set of the implicit function: function equations are solvable if the Jacobian matrix with respect to certain variables is locally invertible.

Generalizations and related areas

• In many applications it is desirable to be able to form derivatives for continuous or even discontinuous functions. For example, be modeled by a partial differential equation on the beach a breaking wave, but the function of the height of the wave is not even continuous. For this purpose, generalized mid-20th century the dissipation term in the space of distributions defined and there a weak derivative. Closely related to this is the notion of Sobolev space.
• In differential geometry curved surfaces are examined. For this purpose, the notion of differential form is required.
• The term of the derivative as a linearization can be analogously applied functions between two normalizable topological vector spaces and transmitted ( see main article: Fréchet derivative, Gâteaux differential, Lorch - derivative): ie then in Fréchet differentiable if a continuous linear operator exists, so that

• A transfer of the concept of derivative to rings other than, and ( and algebras over) is the derivation.
• The calculus of finite differences transmits the differential calculus on ranks.