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

EXAMPLE.

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

Let be
are differentiable, then by definition

In addition, as required to prove.

EXAMPLES.

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

Differentiation of a function defined parametrically

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

EXAMPLE. Plot a function
.

y About 1 x

The constructed curve is called cycloid(Fig. 25) and is the trajectory of a point on a circle of radius 1 that rolls without slip along the OX axis.

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

EXAMPLES.
are the parametric equations of the circle, since, obviously,

are the parametric equations of the ellipse, since

are the parametric equations of the parabola

Find the derivative of a function given parametrically:

The derivative of a function defined parametrically is also a function defined parametrically: .

DEFINITION. The second derivative of a function is called the derivative of its first derivative.

derivative -th order is the derivative of its order derivative
.

denote the derivatives of the second and th order like this:

It follows from the definition of the second derivative and the rule of differentiation of a parametrically given function that
To calculate the third derivative, it is necessary to represent the second derivative in the form
and use the resulting rule again. Higher order derivatives are calculated in a similar way.

EXAMPLE. Find first and second order derivatives of a function

.

Basic theorems of differential calculus

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

PROOF. Let be
, for example, is the minimum point. By definition of a minimum point, there is a neighborhood of this point
, within which
, i.e
- increment
at the point
. A-priory
Calculate one-sided derivatives at a point
:

by the passage to the limit theorem in the inequality,

as

, as
But by condition
exists, so the left derivative is equal to the right one, and this is possible only if

The assumption that
- the maximum point, leads to the same.

The geometric meaning of the theorem:

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

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

1. Let
, then

2. Let
As
either
, or
reached at the extreme point
, but by Fermat's theorem
Q.E.D.

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

The geometric meaning of the theorem:

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

PROOF. Through points A
and B
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 has been proven.

COMMENT. The formula is called Lagrange formula.

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

PROOF. Let us show that
. If
, then the function
would satisfy the condition of Rolle's theorem, so there would be a point
such that
is a contradiction to the condition. Means,
, and both parts of the formula are defined. Let's consider an auxiliary function.

continuous
, differentiable
and
, i.e
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(Theorem L'Hopital-Bernoulli). Let the functions
continuous
, differentiable
,
and
. In addition, there is a finite or infinite
.

Then there is

PROOF. Since according to the condition
, then we define
at the point
, assuming
Then
become continuous
. Let us show that

Let's pretend that
then there is
such that
, since the function
on the
satisfies the conditions of Rolle's theorem. But by condition
- a contradiction. So

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

,
.

Hence we have:
, because if
, then
.

Renaming the variable in the last limit, we obtain the required:

NOTE 1. L'Hopital's rule remains valid even when
and
. It allows you to reveal not only the uncertainty of the form , but also of the form :

.

NOTE 2. If, after applying the L'Hopital rule, the uncertainty is not revealed, then it should be applied again.

EXAMPLE.

COMMENT 3 . L'Hopital's rule is a universal way to reveal uncertainties, but there are limits that can be revealed by applying 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 higher powers

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 a linear function of the argument Dx and is called the differential of the function y \u003d f (x). If f "(x) \u003d 0, then the differential function by definition is considered to be 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 us find out what is the geometric meaning of the 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

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. 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) (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. Hence,