Along with derivatives their functions differentials are one of the basic concepts of differential calculus, the main section of mathematical analysis. Being inextricably interconnected, both of them have been actively used for several centuries to solve almost all the problems that arose in the process of scientific and technological activity of man.
The emergence of the concept of differential
For the first time he explained what a differential is, one of the founders (along with Isaac Newton) of differential calculus, the famous German mathematician Gottfried Wilhelm Leibniz. Before that, mathematicians 17 tbsp. we used a very fuzzy and vague idea of some infinitesimal "indivisible" part of any known function, which represented a very small constant value, but not equal to zero, less than which the values of the function simply can not. Hence, there was only one step before introducing the idea of infinitesimal increments of the arguments of functions and the corresponding increments of the functions themselves, expressed through derivatives of the latter. And this step was taken almost simultaneously by the two aforementioned great scientists.
Based on the need to solve the urgent practical problems of mechanics posed by a rapidly developing industry and technology, Newton and Leibniz created common ways to find the rate of change of functions (primarily with respect to the mechanical speed of a body along a known path), which led to the introduction of such concepts, as the derivative and the differential of the function, and also found an algorithm for solving the inverse problem, how to find the distance traveled by the known (variable) speed, which led to the emergence of the concept of integral ala.
In the writings of Leibniz and Newton, for the first time, the idea appeared that differentials are the main parts of the increments of the functions Δ that are proportional to the increments of the arguments Δx, which can be successfully applied to calculate the values of the latter. In other words, they discovered that the increment of a function can be expressed at any point (within the domain of its definition) through its derivative as Δ = y '(x) Δ + αΔ, where α Δ is the residual term tending to zero as Δ → 0, much faster than Δx itself.
According to the founders of matanalysis, differentials are precisely the first terms in the expressions of increments of any functions. Still not possessing a clearly formulated concept of the limit of sequences, they intuitively realized that the differential value tends to the derivative of the function as Δ → 0 - Δ / Δ → y '(x).
Unlike Newton, who was primarily a physicist, and considered the mathematical apparatus as an auxiliary tool for studying physical problems, Leibniz paid more attention to this toolkit itself, including a system of clear and understandable notation of mathematical quantities. It was he who proposed the generally accepted notation for the differentials of the function dy = y '(x) dx, the argument dx and the derivative of the function in the form of their ratio y' (x) = dy / dx.
Modern definition
What is a differential in terms of modern mathematics? It is closely related to the concept of variable increment. If the variable y takes first the value y = y 1 , and then y = y 2 , then the difference y 2 ─ y 1 is called the increment of y.
The increment may be positive. negative and equal to zero. The word "increment" is denoted by Δ, the record Δy (read "delta igrek") means the increment of y. so that Δy = y
2 ─ y
1 .
If the quantity Δy of an arbitrary function y = f (x) can be represented as Δy = A Δx + α, where A has no dependence on Δx, i.e., A = const for a given x, and the term α as Δx → 0 tends to it is even faster than Δx itself, then the first ("main") term proportional to Δx is for y = f (x) the differential, denoted by dy or df (x) (read “de igrek”, “de eff from x”). Therefore, differentials are the "main" linear components of the increments of functions with respect to Δ.
Mechanical interpretation
Let s = f (t) be the distance of a rectilinearly moving material point from the initial position (t is the travel time). The increment Δs is the path of the point over the time interval Δt, and the differential ds = f '(t) Δt is the path that the point would travel in the same time Δt if it retained the speed f' (t) achieved at time t . For infinitely small Δt, the imaginary path ds differs from the true Δs by an infinitely small value having a higher order relative to Δt. If the velocity at time t is not equal to zero, then ds gives an approximate value of the small displacement of the point.
Geometric interpretation
Let line L be a graph of y = f (x). Then Δ x = MQ, Δy = QM '(see the figure below). The tangent MN splits the segment Δy into two parts, QN and NM '. The first is proportional to Δx and is equal to QN = MQ ∙ tg (angle QMN) = Δ f '(x), i.e., QN is the differential dy.
The second part of NM gives the difference Δy ─ dy, as Δx → 0 the length of NM 'decreases even faster than the increment of the argument, that is, its order of smallness is higher than that of Δx. In the case under consideration, for f '(x) ≠ 0 (the tangent is not parallel to OX), the segments QM'and QN are equivalent; in other words, NM 'decreases faster (its order of smallness is higher) than the total increment Δ = QM'. This can be seen in the figure (with approaching M'k M, the NM segment makes up an ever smaller percentage of the QM 'segment).
So, graphically, the differential of an arbitrary function is equal to the increment of the ordinate of its tangent.
Derivative and differential
The coefficient A in the first term of the expression of the increment of the function is equal to the value of its derivative f '(x). Thus, the following relation holds - dy = f '(x) Δ, or df (x) = f' (x) Δ.
It is known that the increment of an independent argument is equal to its differential Δx = dx. Accordingly, we can write: f '(x) dx = dy.
Finding (sometimes they say “solution”) of differentials is performed according to the same rules as for derivatives. A list of them is given below.
What is more universal: argument increment or its differential
Some clarification is needed here. Representation by the quantity f '(x) Δx of the differential is possible when considering x as an argument. But the function can be complex, in which x can be a function of some argument t. Then the representation of the differential by the expression f '(x) Δx, as a rule, is impossible; except for the case of a linear dependence x = at + b.
As for the formula f '(x) dx = dy, then in the case of an independent argument x (then dx = Δx), and in the case of a parametric dependence of x on t, it represents a differential.
For example, the expression 2 x Δx represents, for y = x 2, its differential when x is an argument. Now put x = t 2 and assume that t is an argument. Then y = x 2 = t 4 .
Then follows (t + Δt) 2 = t 2 + 2tΔt + Δt 2 . Hence, Δx = 2tΔt + Δt 2 . So: 2xΔx = 2t 2 (2tΔt + Δt 2 ).
This expression is not proportional to Δt and therefore now 2xΔx is not a differential. It can be found from the equation y = x 2 = t 4 . It turns out to be equal to dy = 4t 3 Δt.
If we take the expression 2xdx, then it represents the differential y = x 2 for any argument t. Indeed, for x = t 2 we get dx = 2tΔt.
Therefore, 2xdx = 2t 2 2tΔt = 4t 3 Δt, i.e., the expressions of the differentials written through two different variables coincided.
Replacing increments with differentials
If f '(x) ≠ 0, then Δy and dy are equivalent (as Δ → 0); for f '(x) = 0 (which means dy = 0), they are not equivalent.
For example, if y = x 2 , then Δy = (x + Δx) 2 ─ x 2 = 2xΔx + Δx 2 , and dy = 2xΔx. If x = 3, then we have Δy = 6Δx + Δx 2 and dy = 6Δx, which are equivalent due to Δx 2 → 0, for x = 0 the quantities Δy = Δx 2 and dy = 0 are not equivalent.
This fact, together with the simple structure of the differential (i.e., linearity with respect to Δx), is often used in approximate calculations, under the assumption that Δy ≈ dy for small Δx. Finding the differential of a function is usually easier than calculating the exact value of the increment.
For example, we have a metal cube with an edge x = 10.00 cm. When heated, the edge was elongated by Δx = 0.001 cm. How much did the volume of the V cube increase? We have V = x 2 , so dV = 3x 2 Δx = 3 ∙ 10 2 ∙ 0/01 = 3 (cm 3 ). The increase in volume ΔV is equivalent to the differential dV, so ΔV = 3 cm 3 . A complete calculation would give ΔV = 10.01 3 ─ 10 3 = 3.003001. But in this result, all the numbers except the first are unreliable; it means, anyway, you need to round it to 3 cm 3 .
Obviously, this approach is useful only if it is possible to estimate the magnitude of the introduced error.
Function Differential: Examples
Let us try to find the differential of the function y = x 3 without finding the derivative. We give the argument an increment and determine Δy.
Δ = (Δ + x) 3 ─ x 3 = 3x 2 Δ + (3xΔ 2 + Δ 3 ).
Here the coefficient A = 3x 2 is independent of Δx, so the first term is proportional to Δx, while the other term is 3xΔx 2 + Δx 3 as Δ → 0 decreases faster than the increment of the argument. Therefore, the term 3x 2 Δx is the differential y = x 3:
dy = 3x 2 Δx = 3x 2 dx or d (x 3 ) = 3x 2 dx.
Moreover, d (x 3 ) / dx = 3x 2 .
We now find the dy of the function y = 1 / x through its derivative. Then d (1 / x) / dx = ─1 / x 2 . Therefore, dy = ─ Δx / x 2 .
The differentials of the basic algebraic functions are given below.
Approximate differential calculations
It is often not difficult to calculate the function f (x), as well as its derivative f '(x) for x = a, but it can be difficult to do the same in the vicinity of the point x = a. Then an approximate expression comes to the rescue
f (a + Δ) ≈ f '(a) Δ + f (a).
It gives an approximate value of the function at small increments Δx through its differential f '(a) Δ.
Therefore, this formula gives an approximate expression for the function at the end point of a certain section of length Δx as the sum of its value at the starting point of this section (x = a) and the differential at the same starting point. The error of this method of determining the value of the function is illustrated in the figure below.
However, the exact expression of the value of the function for x = a + Δx is also given by the formula of finite increments (or, otherwise, the Lagrange formula)
f (a + Δ) ≈ f '(ξ) Δ + f (a),
where the point x = a + ξ is in the interval from x = a to x = a + Δ, although its exact position is unknown. The exact formula allows us to estimate the error of the approximate formula. If we put ξ = Δ / 2 in the Lagrange formula, then although it ceases to be exact, it usually gives a much better approximation than the original expression through the differential.
Error estimation of formulas using differential
Measuring instruments are inaccurate in principle, and introduce corresponding errors into the measurement data. They are characterized by a marginal absolute error, or, in short, a marginal error - a positive number that obviously exceeds this error in absolute value (or in extreme cases equal to it). The limiting relative error is the quotient of its division by the absolute value of the measured quantity.
Suppose that the exact formula y = f (x) is used to calculate the function y, but the value x is the measurement result and therefore introduces an error in y. Then, to find the limiting absolute error │Δy of function y, use the formula
│Δ│≈│dy│ = │ f '(x) ││Δ│,
where │Δ is the marginal error of the argument. The value of │Δ│ should be rounded up; it is inaccurate to replace the calculation of the increment with the calculation of the differential.