Differentials are what? How to find the differential of a function?

Along with the derivatives Their functions Differentials are One of the basic concepts of differential calculus, the main section of mathematical analysis. Being inextricably linked, they have been actively used for several centuries to solve almost all the problems that arose in the process of scientific and technical activities of man.

The origin of the notion of differential

For the first time he explained what a differential is, one of the creators (along with Isaac Newton) of the differential calculus, the famous German mathematician Gottfried Wilhelm Leibniz. Prior to this mathematicians 17 art. A very vague and vague idea was used about some infinitely small "indivisible" part of any known function that 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 the introduction of the notion of infinitesimal increments in the arguments of functions and the corresponding increments of the functions themselves, expressed in terms of the derivatives of the latter. And this step was made almost simultaneously by the two above-mentioned great scientists.

Proceeding from the need to solve the pressing practical problems of mechanics, which posed rapidly developing industry and technology to science, Newton and Leibniz created common methods for finding the rate of change of functions (primarily with respect to the mechanical velocity of the body along a known trajectory), which led to the introduction of such concepts, As the derivative and differential of a function, and also found the algorithm for solving the inverse problem, as in the known (variable) velocity find the path traversed, which led to the appearance of the concept of integr Ala.

In the writings of Leibniz and Newton, it first appeared that the differentials are proportional to the increments of the arguments Δx, the main parts of the increments of the functions Δy, 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 (inside the domain of its definition) through its derivative as Δy = y '(x) Δx + αΔx, where αΔx is the remainder tending to zero as Δx → 0, is much faster than Δx itself.

According to the founders of the matanalysis, the differentials are precisely the first terms in the expressions for the increments of any functions. Not yet having a clearly defined concept of the limit of sequences, they intuitively understood that the value of the differential tends to the derivative of the function as Δx → 0 - Δy / Δx → 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 instrument itself, including a system of intuitive and understandable designations for 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 the differential in terms of modern mathematics? It is closely related to the notion of increment of a variable. If the variable y takes the value y = y _{1 first} , and then y = y ₂ , then the difference y ₂ ─ y ₁ is called the increment of y. The increment can be positive. Negative and equal to zero. The word "increment" is denoted by Δ, the record Δy (read "delta yerk") denotes the increment of y. So that Δy = y ₂ ─ y ₁ .

If the value Δy of an arbitrary function y = f (x) can be represented as Δy = A Δx + α, where A does not depend on Δx, that is, 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 ("principal") term proportional to Δx, and is for y = f (x) the differential denoted by Dy or df (x) (it reads "de game", "de eff from x"). Therefore, differentials are the "main" components of the increments of functions that are linear with respect to Δx.

Mechanical interpretation

Let s = f (t) be the distance of a rectilinearly moving material point from the initial position (t is the time spent in the path). 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 have passed in the same time Δt if it had retained the speed f' (t) reached at time t . For an infinitesimal Δt, the imaginary path ds differs from the true Δs by an infinitesimal amount having a higher order with respect to Δt. If the velocity at time t is not zero, then ds gives an approximate value of the small displacement of the point.

Geometric interpretation

Let the line L be the graph of y = f (x). Then Δx = MQ, Δy = QM '(see the figure below). The tangent MN divides the segment Δy into two parts, QN and NM '. The first is proportional to Δx and is equal to QN = MQ ∙ tg (angle QMN) = Δx f '(x), that is, QN is the differential dy.

The second part of the NM is given by the difference Δy ─ dy; for Δx → 0, the length of NM 'decreases even more rapidly 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 (tangent is not parallel to OX), the segments QM'and QN are equivalent; In other words, NM 'decreases faster (the order of smallness is higher) than the total increment Δy = QM'. This is seen in the figure (with the approach of M'kM, the segment NM'is the ever smaller percentage of the segment QM ').

Thus, graphically the differential of an arbitrary function is equal to the magnitude of the increment of the ordinate of its tangent.

Derivative and differential

The coefficient A in the first term of the expression for the increment of a function is equal to its derivative f '(x). Thus, the following relation holds: dy = f '(x) Δx, or df (x) = f' (x) Δx.

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.

The finding (sometimes say, the "solution") of differentials is fulfilled by the same rules as for the derivatives. The list of them is given below.

What is more universal: the increment of the argument or its differential

Here it is necessary to make some explanations. A representation of the differential f '(x) Δx is possible when x is considered 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 the 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 its differential for y = x ² , when x is an argument. We now set x = t ² and consider t an argument. Then y = x ² = t ⁴ .

Then follows (t + Δt) ² = t ² + 2tΔt + Δt ² . Hence Δx = 2tΔt + Δt ² . Hence: 2xΔx = 2t ² (2tΔt + Δt ² ).

This expression is not proportional to Δt and therefore 2xΔx is not a differential. It can be found from the equation y = x ² = t ⁴ . It turns out to be 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 obtain dx = 2tΔt.

Hence 2xdx = 2t ² 2tΔt = 4t ³ Δt, that is, the expressions for the differentials written through two different variables coincide.

Replacement of increments by differentials

If f '(x) ≠ 0, then Δy and dy are equivalent (for Δx → 0); For f '(x) = 0 (which means dy = 0), they are not equivalent.

For example, if y = x ² , then Δу = (x + Δх) ² ─ x ² = 2xΔх + Δх ² , and dy = 2xΔх. If x = 3, then we have Δy = 6Δx + Δx ² and dy = 6Δx, which are equivalent due to Δx ² → 0, at x = 0 the values 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 elongated by Δx = 0.001 cm. How much has the volume V of the cube increased? We have V = x ² , so that dV = 3x ² Δх = 3 ∙ 10 ² ∙ 0/01 = 3 (cm ³ ). The increase in volume ΔV is equivalent to the differential dV, so that ΔV = 3 cm ³ . A full calculation would give ΔV = 10.01 ³ ─ 10 ³ = 3.003001. But in this result, all the figures except the first are unreliable; Then, anyway, you need to round it to 3 cm ³ .

Obviously, such an approach is useful only if it is possible to estimate the magnitude of the error that is introduced.

Differential of function: examples

Let's try to find the differential of the function y = x ³ , without finding a derivative. Let's give the argument an increment and define Δy.

Δу = (Δх + x) ³ ─ x ³ = 3x ² Δх + (3xΔх ² + Δх ³ ).

Here the coefficient A = 3x ² does not depend on Δx, so the first term is proportional to Δx, the other term is 3xΔx ² + Δx ³ When Δx → 0, it decreases faster than the increment of the argument. Therefore, the term 3x ² Δx is the differential y = x ^3:

Dy = 3x2 Δx = 3x2 dx or d (x3) = 3x2 dx.

In this case, d (x ³ ) / dx = 3x2.

We now find dy of the function y = 1 / x in terms of 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 computations using the differential

It is often not difficult to compute the function f (x), as well as its derivative f '(x) for x = a, but it is not easy to do the same thing in the neighborhood of the point x = a. Then an approximate expression comes to the rescue

F (a + Δx) ≈ f '(a) Δx + f (a).

It gives an approximate value of the function for small increments Δx through its differential f '(a) Δx.

Consequently, this formula gives an approximate expression for the function at the end point of a section of length Δx as the sum of its value at the start point of this section (x = a) and the differential at the same starting point. The error in this way of determining the value of the function is illustrated in the figure below.

However, an exact expression for the value of the function for x = a + Δx given by the formula of finite increments (or, in other words, by the Lagrange formula)

F (a + Δx) ≈ f '(ξ) Δx + f (a),

Where the point x = a + ξ is on the segment from x = a to x = a + Δx, although its exact position is unknown. The exact formula makes it possible to estimate the error of the approximate formula. If, however, we set ξ = Δx / 2 in the Lagrangian formula, although it ceases to be exact, it usually gives a much better approximation than the original expression in terms of the differential.

Estimation of the error of formulas using the differential

Measuring instruments are in principle inaccurate, and introduce errors into the measurement data. They are characterized by the ultimate absolute error, or, more briefly, by the marginal error-a positive number, which certainly exceeds this error in absolute value (or at most equal to it). The limiting relative error is the quotient of its division by the absolute value of the measured value.

Let the exact formula y = f (x) be used to calculate the function y, but the value of x is the result of the measurement and therefore introduces an error in y. Then, in order to find the limiting absolute error │Δy│ of the function y, use the formula

│Δу│≈│dy│ = │ f '(x) ││Δх│,

Where │Δx│ is the limiting error of the argument. The value of │Δy│ should be rounded upwards. The replacement of the calculation of the increment by the calculation of the differential is inaccurate.