A branch of mathematics dealing with the concepts of derivative and differential and the manner of using them in the study of functions. The development of differential calculus is closely connected with that of integral calculus. Indissoluble is also their content. Together they form the base of mathematical analysis, which is extremely important in the natural sciences and in technology. The introduction of variable magnitudes into mathematics by R. Descartes was the principal factor in the creation of differential calculus. Differential and integral calculus were created, in general terms, by I. Newton and G. Leibniz towards the end of the 17th century, but their justification by the concept of limit was only developed in the work of A.L. Cauchy in the early 19th century. The creation of differential and integral calculus initiated a period of rapid development in mathematics and in related applied disciplines. Differential calculus is usually understood to mean classical differential calculus, which deals with real-valued functions of one or more real variables, but its modern definition may also include differential calculus in abstract spaces. Differential calculus is based on the concepts of real number; function; limit and continuity — highly important mathematical concepts, which were formulated and assigned their modern content during the development of mathematical analysis and during studies of its foundations. The central concepts of differential calculus — the derivative and the differential — and the apparatus developed in this connection furnish tools for the study of functions which locally look like linear functions or polynomials, and it is in fact such functions which are of interest, more than other functions, in applications.

Derivative.

Let a function be defined in some neighbourhood of a point . Let denote the increment of the argument and let denote the corresponding increment of the value of the function. If there exists a (finite or infinite) limit then this limit is said to be the derivative of the function at ; it is denoted by , , , , . Thus, by definition, The operation of calculating the derivative is called differentiation. If is finite, the function is called differentiable at the point . A function which is differentiable at each point of some interval is called differentiable in the interval.

Geometric interpretation of the derivative.

Let be the plane curve defined in an orthogonal coordinate system by the equation where is defined and is continuous in some interval ; let be a fixed point on , let () be an arbitrary point of the curve and let be the secant (Fig. a). An oriented straight line ( a variable point with abscissa ) is called the tangent to the curve at the point if the angle between the secant and the oriented straight line tends to zero as (in other words, as the point arbitrarily tends to the point ). If such a tangent exists, it is unique. Putting , , one obtains the equation for the angle between and the positive direction of the -axis (Fig. a).

Figure: d031850a

Figure: d031850b

Mechanical interpretation of the derivative.

Let a point move in a straight line in accordance with the law . During time the point becomes displaced by . The ratio represents the average velocity during the time . If the motion is non-uniform, is not constant. The instantaneous velocity at the moment is the limit of the average velocity as , i.e. (on the assumption that this derivative in fact exists).

Thus, the concept of derivative constitutes the general solution of the problem of constructing tangents to plane curves, and of the problem of calculating the velocity of a rectilinear motion. These two problems served as the main motivation for formulating the concept of derivative.

A function which has a finite derivative at a point is continuous at this point. A continuous function need not have a finite nor an infinite derivative. There exist continuous functions having no derivative at any point of their domain of definition.

The formulas given below are valid for the derivatives of the fundamental elementary functions at any point of their domain of definition (exceptions are stated):

1) if , then ;

2) if , then ;

3) , (, if );

4) , , ; in particular, ;

5) , , , ;

6) ;

7) ;

8) ;

9) ;

10) , ;

11) , ;

12) ;

13) ;

14) ;

15) ;

16) ;

17) .

The following laws of differentiation are valid:

If two functions and are differentiable at a point , then the functions are also differentiable at that point, and

Theorem on the derivative of a composite function: If the function is differentiable at a point , while the function is differentiable at a point , and if , then the composite function is differentiable at , and or, using another notation, .

Theorem on the derivative of the inverse function: If and are two mutually inverse increasing (or decreasing) functions, defined on certain intervals, and if exists (i.e. is not infinite), then at the point the derivative exists, or, in a different notation, . This theorem may be extended: If the other conditions hold and if also or , then, respectively, or .

One-sided derivatives.

If at a point the limit exists, it is called the right-hand derivative of the function at (in such a case the function need not be defined everywhere in a certain neighbourhood of the point ; this requirement may then be restricted to ). The left-hand derivative is defined in the same way, as: A function has a derivative at a point if and only if equal right-hand and left-hand derivatives exist at that point. If the function is continuous, the existence of a right-hand (left-hand) derivative at a point is equivalent to the existence, at the corresponding point of its graph, of a right (left) one-sided semi-tangent with slope equal to the value of this one-sided derivative. Points at which the semi-tangents do not form a straight line are called angular points or cusps (cf. Fig. c).

Figure: d031850c

Derivatives of higher orders.

Let a function have a finite derivative at all points of some interval; this derivative is also known as the first derivative, or the derivative of the first order, which, being a function of , may in its turn have a derivative , known as the second derivative, or the derivative of the second order, of the function , etc. In general, the -th derivative, or the derivative of order , is defined by induction by the equation , on the assumption that is defined on some interval. The notations employed along with are , , and, if , also , , , .

The second derivative has a mechanical interpretation: It is the acceleration of a point in rectilinear motion according to the law .

Differential.

Let a function be defined in some neighbourhood of a point and let there exist a number such that the increment may be represented as with as . The term in this sum is denoted by the symbol or and is named the differential of the function (with respect to the variable ) at . The differential is the principal linear part of increment of the function (its geometrical expression is the segment in Fig. a, where is the tangent to at the point under consideration).

The function has a differential at if and only if it has a finite derivative at this point. A function for which a differential exists is called differentiable at the point in question. Thus, the differentiability of a function implies the existence of both the differential and the finite derivative, and . For the independent variable one puts , and one may accordingly write , i.e. the derivative is equal to the ratio of the differentials: See also Differential.

The formulas and the rules for computing derivatives lead to corresponding formulas and rules for calculating differentials. In particular, the theorem on the differential of a composite function is valid: If a function is differentiable at a point , while a function is differentiable at a point and , then the composite function is differentiable at the point and , where . The differential of a composite function has exactly the form it would have if the variable were an independent variable. This property is known the invariance of the form of the differential. However, if is an independent variable, is an arbitrary increment, but if is a function, is the differential of this function which, in general, is not identical with its increment.

Differentials of higher orders.

The differential is also known as the first differential, or differential of the first order. Let have a differential at each point of some interval. Here is some number independent of and one may say, therefore, that . The differential is a function of alone, and may in turn have a differential, known as the second differential, or the differential of the second order, of , etc. In general, the -th differential, or the differential of order , is defined by induction by the equality , on the assumption that the differential is defined on some interval and that the value of is identical at all steps. The invariance condition for is generally not satisfied (with the exception where is a linear function).

The repeated differential of has the form and the value of for is the second differential.

Principal theorems and applications of differential calculus.

The fundamental theorems of differential calculus for functions of a single variable are usually considered to include the Rolle theorem, the Legendre theorem (on finite variation), the Cauchy theorem, and the Taylor formula. These theorems underlie the most important applications of differential calculus to the study of properties of functions — such as increasing and decreasing functions, convex and concave graphs, finding the extrema, points of inflection, and the asymptotes of a graph (cf. Extremum; Point of inflection; Asymptote). Differential calculus makes it possible to compute the limits of a function in many cases when this is not feasible by the simplest limit theorems (cf. Indefinite limits and expressions, evaluations of). Differential calculus is extensively applied in many fields of mathematics, in particular in geometry.

Differential calculus of functions in several variables.

For the sake of simplicity the case of functions in two variables (with certain exceptions) is considered below, but all relevant concepts are readily extended to functions in three or more variables. Let a function be given in a certain neighbourhood of a point and let the value be fixed. will then be a function of alone. If it has a derivative with respect to at , this derivative is called the partial derivative of with respect to at ; it is denoted by , , , , , or . Thus, by definition, where is the partial increment of the function with respect to (in the general case, must not be regarded as a fraction; is the symbol of an operation).

The partial derivative with respect to is defined in a similar manner: where is the partial increment of the function with respect to . Other notations include , , , , and . Partial derivatives are calculated according to the rules of differentiation of functions of a single variable (in computing one assumes while if is calculated, one assumes ).

The partial differentials of at are, respectively, where, as in the case of a single variable, , denote the increments of the independent variables.

The first partial derivatives and , or the partial derivatives of the first order, are functions of and , and may in their turn have partial derivatives with respect to and . These are named, with respect to the function , the partial derivatives of the second order, or second partial derivatives. It is assumed that The following notations are also used instead of : and instead of : etc. One can introduce in the same manner partial derivatives of the third and higher orders, together with the respective notations: means that the function is to be differentiated times with respect to ; where means that the function is differentiated times with respect to and times with respect to . The partial derivatives of second and higher orders obtained by differentiation with respect to different variables are known as mixed partial derivatives.

To each partial derivative corresponds some partial differential, obtained by its multiplication by the differentials of the independent variables taken to the powers equal to the number of differentiations with respect to the respective variable. In this way one obtains the -th partial differentials, or the partial differentials of order : The following important theorem on derivatives is valid: If, in a certain neighbourhood of a point , a function has mixed partial derivatives and , and if these derivatives are continuous at the point , then they coincide at this point.

A function is called differentiable at a point with respect to both variables and if it is defined in some neighbourhood of this point, and if its total increment may be represented in the form where and are certain numbers and for (provided that the point lies in this neighbourhood). In this context, the expression is called the total differential (of the first order) of at ; this is the principal linear part of increment. A function which is differentiable at a point is continuous at that point (the converse proposition is not always true!). Moreover, differentiability entails the existence of finite partial derivatives Thus, for a function which is differentiable at , or if, as in the case of a single variable, one puts, for the independent variables, , .

The existence of finite partial derivatives does not, in the general case, entail differentiability (unlike in the case of functions in a single variable). The following is a sufficient criterion of the differentiability of a function in two variables: If, in a certain neighbourhood of a point , a function has finite partial derivatives and which are continuous at , then is differentiable at this point. Geometrically, the total differential is the increment of the applicate of the tangent plane to the surface at the point , where (cf. Fig. d).

Figure: d031850d

Total differentials of higher orders are, as in the case of functions of one variable, introduced by induction, by the equation on the assumption that the differential is defined in some neighbourhood of the point under consideration, and that equal increments of the arguments , are taken at all steps. Repeated differentials are defined in a similar manner.

Derivatives and differentials of composite functions.

Let be a function in variables which is differentiable at each point of an open domain of the -dimensional Euclidean space , and let functions in variables be defined in an open domain of the -dimensional Euclidean space . Finally, let the point , corresponding to a point , be contained in . The following theorems then hold:

A) If the functions have finite partial derivatives with respect to , the composite function in also has finite partial derivatives with respect to , and

B) If the functions are differentiable with respect to all variables at a point , then the composite function is also differentiable at that point, and where are the differentials of the functions . Thus, the property of invariance of the first differential also applies to functions in several variables. It does not usually apply to differentials of the second or higher orders.

Differential calculus is also employed in the study of the properties of functions in several variables: finding extrema, the study of functions defined by one or more implicit equations, the theory of surfaces, etc. One of the principal tools for such purposes is the Taylor formula.

The concepts of derivative and differential and their simplest properties, connected with arithmetical operations over functions and superposition of functions, including the property of invariance of the first differential, are extended, practically unchanged, to complex-valued functions in one or more variables, to real-valued and complex-valued vector functions in one or several real variables, and to complex-valued functions and vector functions in one or several complex variables. In functional analysis the ideas of the derivative and the differential are extended to functions of the points in an abstract space.

For the history of differential and integral calculus, see [1]–[6]. For studies by the founders and creators of differential and integral calculus, see [7]–[13]. For handbooks and textbooks of differential and integral calculus, see [14]–[24].

References

See also Gâteaux derivative; Fréchet derivative; Schwarz differential for generalizations. There are many books treating the subject mentioned above. A few are given below.

References