The expression for the total differential of a function of several variables has the same form regardless of 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.Full derivative of a function- derivative of the function with respect to time 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 point) in this case is equal to the partial derivative with respect to time (at the corresponding point) and can be calculated using the formula:

Where - partial derivatives. It should be noted that the designation is conditional and has no relation to 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 the function:

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

Full differential

Full differential

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

in the case where it differs from the full increment

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

by an amount infinitely small compared to

Tangent plane to surface

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

1.

2.

3.

Normal to surface

3.

4.

The concept of differential. Geometric meaning of 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 = f"(x)D x +a (D x) D x,

where the first term is linear with respect to Dx, and the second is infinite at the point Dx = 0 small function more high order, than Dx. If f"(x)№ 0, then the first term represents the main part of the increment Dy. This main part increment is linear function argument Dx and is called the differential of the function y = f(x). If f"(x) = 0, then the differential of the function is considered equal to zero by definition.

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

Note that the differential of the 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 = f"(x)dx. (4)

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

KN = MNtgf = D xtg f = f"(x)D x,

that is, dy = KN.

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

Let us 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 complex function according to Theorem (3) is equal to y" = f"(u) · u". Then the differential of the function

dy = f"(x)dx = f"(u)u"dx = 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 a differential is called the invariance of the form of the first differential.

Comment. Note that in formula (4) dx = Dx, and 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 forms one of the main parts

Function differential

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

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

Where ω (x - x 0) = O(x - x 0) at x → x 0 .

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

For the function differential value f accepted designation df or df(x 0) if you need to know at what point it was calculated. Thus,

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) (at 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 relative to the increment h = x - x 0 .

Criterion for differentiability of a function

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

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

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

By definition, the differential (first differential) of a function is calculated by the formula
If – independent variable.

EXAMPLE.

Let us show that the form of the first differential remains unchanged (is invariant) even in the case when the argument of the function itself is a function, that is, for a complex function
.

Let
are differentiable, then by definition

Moreover, that is what needed to be proven.

EXAMPLES.

The proven invariance of the form of the first differential allows us to assume that
that is the derivative is equal to the ratio of the differential of the function to the differential of her argument, regardless of whether the argument is an independent variable or a function.

Differentiation of a function specified parametrically

Let If function
has on set the opposite, then
Then the equalities
defined on the set function specified parametrically, – parameter (intermediate variable).

EXAMPLE.
.

Graph the function y O 1

x The constructed curve is called cycloid (Fig. 25)

and is the trajectory of a point on a circle of radius 1, which rolls without sliding along the OX axis. COMMENT

EXAMPLES.
– . Sometimes, but not always, a parameter can be eliminated from the parametric curve equations. parametric equations

circles, since obviously

–parametric equations of the ellipse, since

–parametric equations of a parabola

Let's find the derivative of a function defined parametrically: .

The derivative of a function specified parametrically is also a function specified parametrically: DEFINITION

. The second derivative of a function is the derivative of its first derivative. Derivative
.

the th order is the derivative of its derivative of order Denote derivatives of the second and

From the definition of the second derivative and the rule of differentiation of a parametrically defined function it follows that
To calculate the third derivative, you need to represent the second derivative in the form
and use the resulting rule again. Higher order derivatives are calculated similarly.

EXAMPLE. Find the first and second order derivatives of the function

.

Basic theorems of differential calculus

THEOREM(Farm). Let the function
has at point
extremum. If exists
, That

PROOF. Let
, for example, is the minimum point. By definition of a minimum point, there is a neighborhood of this point
, within which
, that is
– increment
at the point
. A-priory
Let's calculate one-sided derivatives at the point
:

by the theorem on passage to the limit in inequality,

because

, because
But according to the condition
exists, therefore the left derivative is equal to the right one, and this is only possible if

The assumption that
– the maximum point leads to the same thing.

Geometric meaning of the theorem:

THEOREM(Rolla). Let the function
continuous
, differentiable
And
then there is
such that

PROOF. Because
continuous
, then by Weierstrass’s second theorem it reaches at
their greatest
and the least
values ​​either at extremum points or at the ends of the segment.

1. Let
, Then

2. Let
Because
either
, or
is reached at the extremum point
, but according to Fermat's theorem
Q.E.D.

THEOREM(Lagrange). Let the function
continuous
and differentiable
, then there is
such that
.

Geometric meaning of the theorem:

Because
, then the secant is parallel to the tangent. Thus, the theorem states that there is a tangent parallel to the secant passing through points A and B.

PROOF. Through points A
and B
Let's draw a secant AB. Her equation
Consider the function

– the distance between the corresponding points on the graph and on the secant AB.

1.
continuous
as the difference of continuous functions.

2.
differentiable
as the difference of differentiable functions.

3.

Means,
satisfies the conditions of Rolle's theorem, so there exists
such that

The theorem is proven.

COMMENT. The formula is called Lagrange's formula.

THEOREM(Cauchy). Let the functions
continuous
, differentiable
And
, then there is a point
such that
.

PROOF. Let's show that
. If
, then the function
would satisfy the conditions of Rolle's theorem, so there would be a point
such that
– a contradiction to the condition. Means,
, and both sides of the formula are defined. Let's look at a helper function.

continuous
, differentiable
And
, that is
satisfies the conditions of Rolle's theorem. Then there is a point
, wherein
, But

Q.E.D.

The proven formula is called Cauchy formula.

L'Hopital's RULE(L'Hopital-Bernoulli theorem). Let the functions
continuous
, differentiable
,
And
. In addition, there is a finite or infinite
.

Then there is

PROOF. Since by condition
, then we define
at the point
, assuming
Then
will become continuous
. Let's show that

Let's pretend that
then there is
such that
, since the function
on
satisfies the conditions of Rolle's theorem. But according to the condition
– a contradiction. That's why

. Functions
satisfy the conditions of Cauchy's theorem on any interval
, which is contained in
. Let's write the Cauchy formula:

,
.

From here we have:
, because if
, That
.

Renaming the variable to the last limit, we get the required:

NOTE 1. L'Hopital's rule remains valid even when
And
. It allows us to reveal not only the uncertainty of the type :

.

, but also the type NOTE 2

EXAMPLE.

and is the trajectory of a point on a circle of radius 1, which rolls without sliding along the OX axis. 3 . If, after applying L'Hopital's rule, the uncertainty is not revealed, then it should be applied again.

. L'Hopital's rule is a universal way of revealing uncertainties, but there are limits that can be revealed by using only one of the previously studied particular techniques.
But obviously

, since the degree of the numerator is equal to the degree of the denominator, and the limit is equal to the ratio of the coefficients at the highest powers
(1),

We have seen that the differential of a function can be written as: If there is an independent variable. Let it now there is a complex function from
,
, i.e.
and therefore
And
. If the derivatives of the functions
exist, then
, as the derivative of a complex function. Differential
or. But
and therefore we can write
, i.e. got the expression for again

as in (1). Conclusion: formula (1) is correct as in the case when there is an independent variable, so is the case when there 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 conservation property (1) is called.

invariance of the differential form

The invariance of the differential form provides great benefits when calculating differentials of complex functions. For example
: need to calculate . Regardless of whether the variable is dependent or independent , we can write it down. If
- function, for example
, then we will find

and, using the invariance of the form of the differential, we have the right to write.

§18. Derivatives of higher orders.

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

the derivative of 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 a second-order derivative is called a third-order derivative or third derivative.

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

In general, the nth order derivative of a function y = (x) is called the derivative of the (n-1) order derivative of this function. (if they exist, of course).

Designate

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

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

In parentheses 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.

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

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

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

2) y=e x; y 1 = e x; y 11 =e x;…;

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

Note that the second derivative has a certain mechanical sense.

If the first derivative of a path with respect to time is the speed of rectilinear non-uniform 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. acceleration of movement:

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

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

In some 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 =a x lna; y 11 =a x (lna) 2; y (n) =a x (lna) n;

y=x α; y 1 = αx α-1; y 11 =
; y (n) = α(α-1)… (α-n+1)x α-n, with =n we have

y (n) = (x n) (n) = n! Derivatives of order higher are all equal to zero.

y=sinx; y 1 = cosх; y 11 = -sinх; y 111 = -cosх; y (4) = sinx;... etc.. Because

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

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

1) (СU) (n) = С(U) (n) ;

2) (U±V) (n) = U (n) ± V (n)

The formula for the nth derivative of the product of two functions (U·V) (n) turns out to be more complex. It is called Leibniz's formula.

y=U·V; y 1 = U 1 V+ UV 1; y 11 = U 11 V+ U 1 V 1 + U 1 V 1 + UV 11 = U 11 V+2U 1 V 1 + UV 11;

y 111 = U 111 V+ U 11 V 1 +2U 11 V 1 +2U 1 V 11 + U 1 V 11 + UV 111 = U 111 V+3U 11 V 1 +3 U 1 V 11 + UV 111;

Similarly we get

y (4) = 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 notice that the right-hand sides of all these formulas resemble the expansion of powers of the binomial U + V, (U + V) 2, (U + V) 3, etc. Only instead of powers 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-к+1)/К! U (k) V (n-k) +…+ UV (n) - Leibniz formula.

Example: find (e x x) (n)

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