The concept of the derivative function at the point. One-sided and endless derivatives. §3. Endless and unilateral derivatives endless derivative function at point

Definition 1. It is said that the functions are at the point x. 0 infinite derivative if

At the same time they write
or
.

Example 1.
,
:

II. One-sided derivatives

Definition 2. Right
and left
derived functions
at point x. 0, determined by the equality:

and
.

From the general theorems of the limits, you can get such a theorem.

Theorem 1. Function
has in point x. 0 Derivative if and only when it has at this point equal to each other one-sided derivatives.

Example 2. For function
find the right and left derivative in zero.

As
T.
does not exist.

The following theorem allows in some cases to simplify the calculation of one-sided derivatives.

Theorem 2. Let the function
has in the interval
eltimate derivative
, moreover, exists (finite or not)
. Then at the point x. 0 There is a right derivative and
.

A similar statement also occurs for the left derivative.

The derivative function calculated in §2
for
:
. Example 1 result (
) With the help of Theorem 2 it turns out instantly:

Similarly, it turns out I.
. Coincidence of unilateral derivatives means that
.

Comment. If function
there are finite, not equal to each other derivatives.
and
then the graph of the function has no coinciding right and left tangents at the point
. Such a point of schedule is called angular. If the derivative (at least one-sided) is + 
This means that the graph has vertical tangent.

§four. Differential function

Definition. It is said that the function
differential in point x. 0, if it is increed to imagine in the form

where A.- some number that does not depend on
.

Theorem 1. In order to function
, was differentiated at the point x. 0, it is necessary and enough for it to have at this point the final derivative.

Evidence. Necessity.Let be
differential. We divide both parts of equality (1) on
:

Moving to the limit when
, get

those. At point x. 0 There is a derivative and it is equal A.:
.

Adequacy. Suppose there is a finite derivative
.

Then
and, therefore,

In this ratio it is not difficult to see equality (1). Theorem is proved.

Thus, for the function of one variable, the differentiability and the existence of a finite derivative - the concepts are equivalent.

they call the formula of infinitely small increments.

There is a link between the concepts of differentiability and continuity, set by the following theorem.

Theorem 2. If the function
differential in point x. 0, then it is continuous at this point.

Indeed from formula (1) it follows that
And this is one of the definitions of continuity.

Naturally, the question of whether the approval is fair to theorem 2, i.e. "Continuous function differentiating." A negative answer should be given to this question: there are functions that are continuous at some point, but not differentiable at this point. An example is the function from Example 2 §3:
. It is continuous in zero, but
does not exist.

Let us give another example of such a function.

Example 1.

This feature is a non-elementary, the possible break point
(At this point, one elementary expression changes to another). But

hence,
continuous at point
. Find a derivative function in zero (by definition!):

But we already know that when the argument of sinus is striving in , the sinus of the limit has no limit. So,
does not exist, i.e.
undifferentiated in zero.

It should be noted that the mathematicians constructed examples of functions continuous at some interval, but not having a derivative at any point of this gap.

Definition 1. It is said that the functions are at the point x. 0 infinite derivative if

At the same time they write
or
.

Example 1.
,
:

II. One-sided derivatives

Definition 2. Right
and left
derived functions
at point x. 0, determined by the equality:

and
.

From the general theorems of the limits, you can get such a theorem.

Theorem 1. Function
has in point x. 0 Derivative if and only when it has at this point equal to each other one-sided derivatives.

Example 2. For function
find the right and left derivative in zero.

As
T.
does not exist.

The following theorem allows in some cases to simplify the calculation of one-sided derivatives.

Theorem 2. Let the function
has in the interval
eltimate derivative
, moreover, exists (finite or not)
. Then at the point x. 0 There is a right derivative and
.

A similar statement also occurs for the left derivative.

The derivative function calculated in §2
for
:
. Example 1 result (
) With the help of Theorem 2 it turns out instantly:

Similarly, it turns out I.
. Coincidence of unilateral derivatives means that
.

§four. Differential function

Definition. It is said that the function
differential in point x. 0, if it is increed to imagine in the form

where A.- some number that does not depend on
.

Theorem 1. In order to function
, was differentiated at the point x. 0, it is necessary and enough for it to have at this point the final derivative.

Evidence. Necessity.Let be
differential. We divide both parts of equality (1) on
:

Moving to the limit when
, get

those. At point x. 0 There is a derivative and it is equal A.:
.

Adequacy. Suppose there is a finite derivative
.

Then
and, therefore,

In this ratio it is not difficult to see equality (1). Theorem is proved.

Thus, for the function of one variable, the differentiability and the existence of a finite derivative - the concepts are equivalent.

they call the formula of infinitely small increments.

There is a link between the concepts of differentiability and continuity, set by the following theorem.

Theorem 2. If the function
differential in point x. 0, then it is continuous at this point.

Indeed from formula (1) it follows that
And this is one of the definitions of continuity.

Let us give another example of such a function.

Example 1.

This feature is a non-elementary, the possible break point
(At this point, one elementary expression changes to another). But

hence,
continuous at point
. Find a derivative function in zero (by definition!):

But we already know that when the argument of sinus is striving in , the sinus of the limit has no limit. So,
does not exist, i.e.
undifferentiated in zero.

It should be noted that the mathematicians constructed examples of functions continuous at some interval, but not having a derivative at any point of this gap.

Let the function f (x) \u003d y be determined in some neighborhood of the point x 0.

Definition 8.1. The derivative function F at point x 0 is called the number indicated equal to the limit of the function of the increment of the function at this point to the increment of the argument Δx when the Δx is to zero if this limit exists:

or, if you designate, then

Definition 8.2. A function having a finite derivative at point X 0 is called differentiable at this point.

Definition 8.3. If in points 0, the function f (x) is continuous, and the limit (8.1) is equal to infinity (+ ∞ or -∞), then they are talking about an infinite derivative.

Definition 8.4. Limits

are called right-hand and left-sided derivative, respectively.

For the existence, the derivative is necessary and enough to have both unilateral derivatives and they were equal to each other:

The derivative is denoted by other methods, for example:

Geometric meaning derivative. The equations of tangent and normal to a flat curve. The angle between the curves.

On the curve f (x) y, we choose two different points M 0 and M 1 (Fig. 8.1) and through them we will carry out the only direct L, which is called the sequential schedule. Using the equations of direct passing through two setpoint and which has the view , we obtain the equation of the section

Comparing equation (8.4) with a direct equation with an angular co-effect, conclude that the angular coefficient K of the section L is

Then and the equation of the secant (8.4) will switch to the equation of tangent:

Thus, the derivative function f (x) \u003d y, calculated at point x \u003d x 0 there is an angular coefficient of tangent, carried out to the graph of the function f (x) \u003d y at the point

This is the geometric meaning of the derivative.

Definition 8.6. Direct, perpendicular to the tangent at point M 0 is called normal to the curve f (x) \u003d y at the point M 0.

From the condition k 1 k 2 \u003d - 1 perpendicularity of the straight lines, we conclude that the angular coefficient K N of normal is expressed through the angular coefficient K CAS tangential according to the formula Consequently, the equation of normal to the curve f (x) \u003d y at the point M 0 has the appearance

Definition 8.7. Let the two curves f (x) \u003d y and g (x) \u003d y intersect at the point ie. The angle α between the specified curves is called an angle between the tangents to the curves carried out at the point of their intersection:

Dock: x \u003d siny

Y / \u003d. 1_ = 1____ =1________

x / Y Cosy COS (ArcSinx)

= 1___________ = 1___

√1-sin 2 * Arccosx √1-x 2

38. Derived reverse function. (with proof)

Let the function y \u003d f (x) (1), is given on the set x (large), and y - the set of its possible values \u200b\u200bthen each x € x is put in line with the only value of the € y, on the other side, each y y will fit one or several values \u200b\u200bx € x. In the case when each in € y does only correspond to one x € x, for which f (x) \u003d y on the set y, it is possible to determine the function x \u003d g (y) (2) of which the set of whose values \u200b\u200bis Many x. The function (2) is called reverse with respect to the 1st. Functions (1) and (2) - converged functions.

Denote in reverse function x \u003d (y).

T.1: If the function y \u003d f (x) is defined strictly monotonously and continuously on the segment, the reverse function x \u003d (y) is defined strictly monotonously and continuously on the segment [A, B], where A \u003d F (a), in \u003d f (b). Strict monotony: for any points, € x< ( > ) Inequality f () f (,))

T.2: Let the function y \u003d f (x) satisfies the conditions of the theorem on the existence of the feed function and at the point has a finite derivative F '() ≠ 0, then the function x \u003d G (y) the point also has a finite derivative of equal.

Proof: Press the increment in ≠ 0, then the function x \u003d g (y) will receive the increment X ≠ 0. Obviously, \u003d.

Definition . The form of the ellipse is determined by the characteristic that is the ratio of the focal length to the larger axis and is called eccentricity. E \u003d C / a.Because from< a, то е < 1.

Two straight lines are associated with ellipse directress. Their equations: x \u003d A / E; x \u003d -a / e.