The rule for differentiating a complex function will lead us to one remarkable and important property of a differential.

Let the functions be such that a complex function can be composed of them: . If there are derivatives, then - by the rule V - there is also a derivative

Replacing, however, its derivative by expression (7) and noticing that there is a differential of x as a function of t, we finally obtain:

i.e., let's return to the previous form of the differential!

Thus, we see that the form of the differential can be preserved even if the old independent variable is replaced by a new one. We are always free to write the differential of y in the form (5), whether x is an independent variable or not; the only difference is that if t is chosen as an independent variable, then it means not an arbitrary increment, but a differential x as a function of This property is called the invariance of the form of the differential.

Since formula (5) directly yields formula (6), which expresses the derivative in terms of differentials, the last formula remains valid, no matter what independent variable (of course, the same in both cases) the named differentials are calculated.

Let, for example, so

We now set Then we will also have: It is easy to check that the formula

gives only another expression for the derivative calculated above.

This circumstance is especially convenient to use in cases where the dependence of y on x is not directly specified, but instead the dependence of both variables x and y on some third, auxiliary, variable (called the parameter) is given:

Assuming that both of these functions have derivatives and that for the first of them there is an inverse function that has a derivative, it is easy to see that then y also turns out to be a function of x:

for which there is also a derivative. The calculation of this derivative can be performed according to the above rule:

without restoring the direct dependence of y on x.

For example, if the derivative can be defined, as it was done above, without using the dependency at all.

If we consider x and y as rectangular coordinates of a point on the plane, then equations (8) assign each value of the parameter t to a certain point, which, with a change in t, describes a curve on the plane. Equations (8) are called parametric equations this curve.

In the case of a parametric specification of the curve, formula (10) makes it possible to establish directly from equations (8) slope tangent without proceeding to specifying the curve by equation (9); exactly,

Comment. The possibility of expressing the derivative in terms of differentials taken with respect to any variable, in particular, leads to the fact that the formulas

expressing in Leibniz notation the rules of differentiation inverse function and a complex function, become simple algebraic identities (since all differentials here can be taken with respect to the same variable). However, one should not think that this gives a new derivation of the above formulas: first of all, the existence of derivatives on the left was not proved here, but the main thing is that we essentially used the invariance of the form of the differential, which itself is a consequence of the rule V.

The expression for the total differential of a function of several variables is the same whether u and v are independent variables or functions of other independent variables.

The proof is based on the total differential formula

Q.E.D.

5.Total derivative of a function is the time derivative of the function along the trajectory. Let the function have the form and its arguments depend on time: . Then , where are the parameters defining the trajectory. The total derivative of the function (at the point ) in this case is equal to the partial time derivative (at the corresponding point ) and can be calculated by the formula:

where - partial derivatives. It should be noted that the designation is conditional and has nothing to do with the division of differentials. In addition, the total derivative of a function depends not only on the function itself, but also on the trajectory.

For example, the total derivative of a function:

There is no here, since in itself (“explicitly”) does not depend on .

Full differential

Full differential

functions f (x, y, z, ...) of several independent variables - expression

in the case when it differs from the full increment

Δf = f(x + Δx, y + Δy, z + Δz,…) - f(x, y, z, …)

to an infinitesimal value compared to

Tangent plane to surface

(X, Y, Z - current coordinates of the point on the tangent plane; - radius vector of this point; x, y, z - coordinates of the tangent point (respectively for the normal); - tangent vectors to the coordinate lines, respectively v = const; u = const ; )

1.

2.

3.

Surface normal

3.

4.

The concept of a differential. The geometric meaning of the differential. Invariance of the form of the first differential.

Consider a function y = f(x) differentiable at a given point x. Its increment Dy can be represented as

D y \u003d f "(x) D x + a (D x) D x,

where the first term is linear with respect to Dx, and the second term at the point Dx = 0 is an infinitesimal function of a higher order than Dx. If f "(x) No. 0, then the first term is the main part of the increment Dy. This main part of the increment is linear function argument Dx and is called the differential of the function y = f(x). If f "(x) \u003d 0, then the differential of the function, by definition, is considered to be equal to zero.

Definition 5 (differential). The differential of the function y = f(x) is the main linear with respect to Dx part of the increment Dy, equal to the product of the derivative and the increment of the independent variable

Note that the differential of an independent variable is equal to the increment of this variable dx = Dx. Therefore, the formula for the differential is usually written in the following form: dy \u003d f "(x) dx. (4)

Let's find out what geometric sense differential. Take an arbitrary point M(x, y) on the graph of the function y = f(x) (Fig. 21.). Draw a tangent to the curve y = f(x) at the point M, which forms an angle f with the positive direction of the axis OX, that is, f "(x) = tgf. From the right triangle MKN

KN \u003d MNtgf \u003d D xtg f \u003d f "(x) D x,

i.e. dy = KN.

Thus, the differential of a function is the increment in the ordinate of the tangent drawn to the graph of the function y = f(x) at a given point when x is incremented by Dx.

We note the main properties of the differential, which are similar to the properties of the derivative.

2. d(c u(x)) = c d u(x);

3. d(u(x) ± v(x)) = d u(x) ± d v(x);

4. d(u(x) v(x)) = v(x)d u(x) + u(x)d v(x);

5. d(u(x) / v(x)) = (v(x) d u(x) - u(x) d v(x)) / v2(x).

Let us point out one more property that the differential has, but the derivative does not. Consider the function y = f(u), where u = f (x), that is, consider the complex function y = f(f(x)). If each of the functions f and f are differentiable, then the derivative of the compound function, according to Theorem (3), is equal to y" = f"(u) u". Then the differential of the function

dy \u003d f "(x) dx \u003d f "(u) u" dx \u003d f "(u) du,

since u "dx = du. That is, dy = f" (u) du. (5)

The last equality means that the differential formula does not change if, instead of a function of x, we consider a function of the variable u. This property of the differential is called the invariance of the form of the first differential.

Comment. Note that in formula (4) dx = Dx, while in formula (5) du is only the linear part of the increment of the function u.

Integral calculus is a branch of mathematics that studies the properties and methods of calculating integrals and their applications. I. and. is closely related to differential calculus and together with it constitutes one of the main parts

We have seen that the differential of a function can be written as:
(1),

if is an independent variable. Let now there is a complex function , i.e.
,
and therefore
. If the derivatives of the functions
and
exist, then
, as a derivative of a complex function. Differential
or. But
and therefore we can write
, i.e. get the expression for
as in (1).

Conclusion: formula (1) is correct as in the case when is an independent variable, and in the case when is a function of the independent variable . In the first case under
is understood as the differential of the independent variable
, in the second - the differential of the function (in this case
, generally speaking). This shape-preserving property (1) is called differential form invariance.

The invariance of the form of the differential gives great benefits when calculating the differentials of complex functions.

For example: to be calculated
. Whether dependent or independent variable , we can write. If a - function, for example
, then we find
and, using the invariance of the form of the differential, we have the right to write down.

§eighteen. Derivatives of higher orders.

Let the function y \u003d  (x) be differentiable on some interval X, (that is, it has a finite derivative y 1 \u003d  1 (x) at each point of this interval). Then  1 (x) is in X itself a function of x. It may turn out that at some points or at all x 1 (x) itself has a derivative, i.e. there is a derivative of the derivative (y 1) 1 \u003d ( 1 (x) 1. In this case, it is called the second derivative or second-order derivative. They are denoted by symbols y 11,  11 (x), d 2 y / dx 2. If necessary emphasize that the derivative is in m.x 0, then write

y 11 / x \u003d x 0 or 11 (x 0) or d 2 y / dx 2 / x \u003d x 0

the derivative of y 1 is called the first order derivative or the first derivative.

So, the second-order derivative is the derivative of the first-order derivative of a function.

Quite similarly, the derivative (where it exists) of the second order derivative is called the third order derivative or the third derivative.

Designate (y 11) 1 \u003d y 111 \u003d 111 (x) \u003d d 3 y / dx 3 \u003d d 3  (x) / dx 3

In general, the derivative of the n-th order of the function y \u003d  (x) is the derivative of the derivative (n-1) of the order of this function. (if they exist, of course).

designate

Read: n-th derivative of y, from  (x); d n y by d x in the n-th.

Fourth, fifth, etc. it is inconvenient to indicate the order with strokes, therefore they write the number in brackets, instead of  v (x) they write  (5) (x).

In brackets, so as not to confuse the nth order of the derivative and the nth degree of the function.

Derivatives of order higher than the first are called derivatives of higher orders.

It follows from the definition itself that to find the nth derivative, you need to find successively all the previous ones from the 1st to the (n-1)th.

Examples: 1) y \u003d x 5; y 1 \u003d 5x 4; y 11 \u003d 20x 3;

y 111 \u003d 60x 2; y (4) =120x; y (5) =120; y (6) =0,…

2) y=e x; y 1 \u003d e x; y 11 \u003d e x; ...;

3) y=sinx; y 1 = cosx; y 11 = -sinx; y 111 = -cosx; y (4) = sinх;…

Note that the second derivative has a certain mechanical meaning.

If the first derivative of the path with respect to time is the speed of non-uniform rectilinear motion

V=ds/dt, where S=f(t) is the equation of motion, then V 1 =dV/dt= d 2 S/dt 2 is the rate of change of speed, i.e. movement acceleration:

a \u003d f 11 (t) \u003d dV / dt \u003d d 2 S / dt 2.

So, the second derivative of the path with respect to time is the acceleration of the movement of the point - this is the mechanical meaning of the second derivative.

In a number of cases, it is possible to write an expression for a derivative of any order, bypassing intermediate ones.

Examples:

y=e x; (y) (n) = (e x) (n) = e x;

y=a x; y 1 \u003d a x lna; y 11 \u003d a x (lna) 2; y (n) = a x (lna) n;

y \u003d x α; y 1 \u003d αx α-1; y 11 =
; y (p) \u003d α (α-1) ... (α-n + 1) x α-n, with =n we have

y (n) = (x n) (n) = n! The order derivatives above n are all zero.

y \u003d sinx; y 1 = cosx; y 11 = -sinx; y 111 = -cosx; y (4) = sinх;… etc.

y 1 \u003d sin (x + /2); y 11 \u003d sin (x + 2 /2); y 111 \u003d sin (x + 3 /2); etc., then y (n) \u003d (sinx) (n) \u003d sin (x + n /2).

It is easy to establish by successive differentiation and general formulas:

1) (CU) (n) = C (U) (n) ; 2) (U ± V) (n) = U (n) ± V (n)

More complicated is the formula for the nth derivative of the product of two functions (U·V) (n) . It is called the Leibniz formula.

Let's get her

y \u003d U V; y 1 \u003d U 1 V + UV 1; y 11 \u003d U 11 V + U 1 V 1 + U 1 V 1 + UV 11 \u003d U 11 V + 2U 1 V 1 + UV 11;

y 111 \u003d U 111 V + U 11 V 1 + 2U 11 V 1 + 2U 1 V 11 + U 1 V 11 + UV 111 \u003d U 111 V + 3U 11 V 1 +3 U 1 V 11 + UV 111;

Similarly, we get

y (4) \u003d U (4) V + 4 U 111 V 1 +6 U 11 V 11 +4 U 1 V 111 + UV (4), etc.

It is easy to see that the right-hand sides of all these formulas resemble the expansion of the powers of the binomial U+V, (U+V) 2 , (U+V) 3 , etc. Only instead of the powers of U and V there are derivatives of the corresponding orders. The similarity will be especially complete if in the resulting formulas we write instead of U and V, U (0) and V (0) , i.e. 0th derivatives of the functions U and V (the functions themselves).

Extending this law to the case of any n, we obtain the general formula

y(n) = (UV)(n) = U(n) V+ n/1! U (n-1) V 1 + n(n-1)/2! U (n-2) V (2) + n(n-1)(n-2)/3! U (n-3) V (3) +…+ n(n-1)…(n-k+1)/K! U (k) V (n-k) + ... + UV (n) - Leibniz formula.

Example: find (e x x) (n)

(e x) (n) \u003d e x, x 1 \u003d 1, x 11 \u003d 0 and x (n) \u003d 0, therefore (e x x) (n) \u003d (e x) (n) x + n / 1 ! (e x) (n-1) x 1 \u003d e x x + ne x \u003d e x (x + n).

Function differential

The function is called differentiable at a point, limiting for the set E, if its increment Δ f(x 0) corresponding to the increment of the argument x, can be represented as

Δ f(x 0) = A(x 0)(x - x 0) + ω (x - x 0), (1)

where ω (x - x 0) = about(x - x 0) at x → x 0 .

Display, called differential functions f at the point x 0 , and the value A(x 0)h - differential value at this point.

For the value of the function differential f accepted designation df or df(x 0) if you want to know at what point it was calculated. In this way,

df(x 0) = A(x 0)h.

Dividing in (1) by x - x 0 and aiming x to x 0 , we get A(x 0) = f"(x 0). Therefore we have

df(x 0) = f"(x 0)h. (2)

Comparing (1) and (2), we see that the value of the differential df(x 0) (when f"(x 0) ≠ 0) is the main part of the function increment f at the point x 0 , linear and homogeneous at the same time with respect to increment h = x - x 0 .

Function differentiability criterion

In order for the function f was differentiable at a given point x 0 , it is necessary and sufficient that it has a finite derivative at this point.

Invariance of the form of the first differential

If a x is an independent variable, then dx = x - x 0 (fixed increment). In this case we have

df(x 0) = f"(x 0)dx. (3)

If a x = φ (t) is a differentiable function, then dx = φ" (t 0)dt. Consequently,