Points on the graph of a differentiable function. Differentiation of functions. Continuity of a function having a derivative. Theorem

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DERIVATIVE– derivative of the function y = f(x), given on a certain interval ( a, b) at point x of this interval is called the limit to which the ratio of the increment of the function tends f at this point to the corresponding increment of the argument when the increment of the argument tends to zero.

The derivative is usually denoted as follows:

Other designations are also widely used:

Instant speed.

Let the point M moves in a straight line. Distance s moving point, counted from some initial position M 0 , depends on time t, i.e. s there is a function of time t: s= f(t). Let at some point in time t moving point M was at a distance s from the starting position M 0, and at some next moment t+D t found herself in a position M 1 - on distance s+D s from the initial position ( see pic.).

Thus, over a period of time D t distance s changed by the amount D s. In this case they say that during the time interval D t magnitude s received increment D s.

The average speed cannot in all cases accurately characterize the speed of movement of a point M at a point in time t. If, for example, the body at the beginning of the interval D t moved very quickly, and at the end very slowly, then the average speed will not be able to reflect the indicated features of the point’s movement and give an idea of ​​​​the true speed of its movement at the moment t. To more accurately express the true speed using the average speed, you need to take a shorter period of time D t. Most fully characterizes the speed of movement of a point at the moment t the limit to which the average speed tends at D t® 0. This limit is called the current speed:

Thus, the speed of movement at a given moment is called the limit of the path increment ratio D s to time increment D t, when the time increment tends to zero. Because

Geometric meaning of the derivative. Tangent to the graph of a function.

The construction of tangent lines is one of those problems that led to the birth of differential calculus. The first published work related to differential calculus, written by Leibniz, was entitled A new method of maxima and minima, as well as tangents, for which neither fractional nor irrational quantities are an obstacle, and a special type of calculus for this.

Let the curve be the graph of the function y =f(x) in a rectangular coordinate system ( cm. rice.).

At some value x function matters y =f(x). These values x And y the point on the curve corresponds M 0(x, y). If the argument x give increment D x, then the new value of the argument x+D x corresponds to the new function value y+ D y = f(x + D x). The corresponding point of the curve will be the point M 1(x+D x,y+D y). If you draw a secant M 0M 1 and denoted by j the angle formed by a transversal with the positive direction of the axis Ox, it is immediately clear from the figure that .

If now D x tends to zero, then the point M 1 moves along the curve, approaching the point M 0, and angle j changes with D x. At Dx® 0 the angle j tends to a certain limit a and the straight line passing through the point M 0 and the component with the positive direction of the x-axis, angle a, will be the desired tangent. Its slope is:

Hence, f´( x) = tga

those. derivative value f´( x) for a given argument value x equals the tangent of the angle formed by the tangent to the graph of the function f(x) at the corresponding point M 0(x,y) with positive axis direction Ox.

Differentiability of functions.

Definition. If the function y = f(x) has a derivative at the point x = x 0, then the function is differentiable at this point.

Continuity of a function having a derivative. Theorem.

If the function y = f(x) is differentiable at some point x = x 0, then it is continuous at this point.

Thus, the function cannot have a derivative at discontinuity points. The opposite conclusion is incorrect, i.e. from the fact that at some point x = x 0 function y = f(x) is continuous does not mean that it is differentiable at this point. For example, the function y = |x| continuous for everyone x(–Ґ x x = 0 has no derivative. At this point there is no tangent to the graph. There is a right tangent and a left one, but they do not coincide.

Some theorems on differentiable functions. Theorem on the roots of the derivative (Rolle's theorem). If the function f(x) is continuous on the segment [a,b], is differentiable at all interior points of this segment and at the ends x = a And x = b goes to zero ( f(a) = f(b) = 0), then inside the segment [ a,b] there is at least one point x= With, a c b, in which the derivative fў( x) goes to zero, i.e. fў( c) = 0.

Finite increment theorem (Lagrange's theorem). If the function f(x) is continuous on the interval [ a, b] and is differentiable at all interior points of this segment, then inside the segment [ a, b] there is at least one point With, a c b that

f(b) – f(a) = fў( c)(b– a).

Theorem on the ratio of the increments of two functions (Cauchy's theorem). If f(x) And g(x) – two functions continuous on the segment [a, b] and differentiable at all interior points of this segment, and gў( x) does not vanish anywhere inside this segment, then inside the segment [ a, b] there is such a point x = With, a c b that

Derivatives of various orders.

Let the function y =f(x) is differentiable on some interval [ a, b]. Derivative values f ў( x), generally speaking, depend on x, i.e. derivative f ў( x) is also a function of x. When differentiating this function, we obtain the so-called second derivative of the function f(x), which is denoted f ўў ( x).

Derivative n- th order of function f(x) is called the (first order) derivative of the derivative n- 1- th and is denoted by the symbol y(n) = (y(n– 1))ў.

Differentials of various orders.

Function differential y = f(x), Where x– independent variable, yes dy = f ў( x)dx, some function from x, but from x only the first factor can depend f ў( x), the second factor ( dx) is the increment of the independent variable x and does not depend on the value of this variable. Because dy there is a function from x, then we can determine the differential of this function. The differential of the differential of a function is called the second differential or second-order differential of this function and is denoted d 2y:

d(dx) = d 2y = f ўў( x)(dx) 2 .

Differential n- of the first order is called the first differential of the differential n- 1- th order:

d n y = d(d n–1y) = f(n)(x)dx(n).

Partial derivative.

If a function depends not on one, but on several arguments x i(i varies from 1 to n,i= 1, 2,… n),f(x 1,x 2,… x n), then in differential calculus the concept of partial derivative is introduced, which characterizes the rate of change of a function of several variables when only one argument changes, for example, x i. 1st order partial derivative with respect to x i is defined as an ordinary derivative, and it is assumed that all arguments except x i, keep constant values. For partial derivatives, the notation is introduced

The 1st order partial derivatives defined in this way (as functions of the same arguments) can, in turn, also have partial derivatives, these are second order partial derivatives, etc. Such derivatives taken from different arguments are called mixed. Continuous mixed derivatives of the same order do not depend on the order of differentiation and are equal to each other.

Anna Chugainova

Derivative functions at a point is called the limit of the ratio of the increment of the function to the increment of the argument, provided that it tends to zero.

Basic rules for finding the derivative

If - and - are differentiable functions at the point , (i.e. functions that have derivatives at the point), then:

4) .

Table of derivatives of basic functions

1. 8.

2. 9.

3. 10.

5. 12.

6. 13.

7.

The rule for differentiating a complex function. If and , i.e. , where and have derivatives, then

Differentiation of a function specified parametrically. Let the dependence of a variable on a variable be specified parametrically by means of the parameter:

Task 3. Find derivatives of these functions.

1)

Solution. Applying rule 2 for finding derivatives and formulas 1 and 2 of the derivative table, we obtain:

Solution. Applying rule 4 for finding derivatives and formulas 1 and 13 of the derivatives table, we obtain:

.

Solution. Applying rule 3 for finding derivatives and formulas 5 and 11 of the derivative table, we obtain:

Solution. Assuming , where , according to the formula for finding the derivative of a complex function, we obtain:

Solution. We have: Then, according to the formula for finding the derivative of a function specified parametrically, we obtain:

4. Higher order derivatives. L'Hopital's rule.

Second order derivative of the function is called the derivative of its derivative, i.e. . The following notations are used for the second derivative: or , or .

The 1st order derivative of the function is called the derivative of its th-order derivative. For the th order derivative, the following notations are used: or , or .

L'Hopital's rule. Let the functions and be differentiable in a neighborhood of the point and the derivative does not vanish. If the functions and are simultaneously either infinitely small or infinitely large at , and there is a limit of the ratio at , then there is also a limit to the ratio at . Moreover

.

The rule also applies when .

Note that in some cases, the disclosure of uncertainties of the type or may require repeated application of L'Hopital's rule.

Type uncertainties, etc. with the help of elementary transformations they can easily be reduced to uncertainties of the form or .

Task 4. Find the limit using L'Hopital's rule.

Solution Here we have uncertainty of the form , because at . Let's apply L'Hopital's rule:

.

After applying L'Hopital's rule, we again obtained uncertainty of the form, because at . Applying L'Hopital's rule again, we get:

.

5. Function study

a) Increasing and decreasing functions

The function is called increasing on the segment , if for any points and from the segment , where , the inequality holds. If a function is continuous on an interval and for , then it increases on the interval.

The function is called decreasing on the segment , if for any points and from the segment , where , the inequality holds. If a function is continuous on an interval and for , then it decreases on the interval.

If a function is only increasing or only decreasing on a given interval, then it is called monotonous on the interval.

b) Extrema of functions

minimum point functions .

If there is a -neighborhood of the point such that for all points from this neighborhood the inequality holds, then the point is called maximum point functions .

The maximum and minimum points of a function are called its extremum points.

The point is called stationary point, if or does not exist.

If there is a -neighborhood of a stationary point such that for and for , then is the maximum point of the function.

If there is a -neighborhood of a stationary point such that for and for , then the -minimum point of the function .

a) Convex direction. Inflection points

convex up on the interval , if it is located below the tangent plotted to the graph of the function at any point in this interval.

A sufficient condition for upward convexity of the graph of a function on an interval is the fulfillment of the inequality for any of the considered intervals.

The graph of a differentiable function is called convex down on the interval , if it is located above the tangent plotted to the graph of the function at any point in this interval.

A sufficient condition for the downward convexity of the graph of a function on an interval is the fulfillment of the inequality for any of the considered intervals.

The point at which the direction of convexity of the graph of a function changes is called inflection point.

A point where or does not exist is the abscissa of an inflection point if the signs to the left and right of it are different.

d) Asymptotes

If the distance from a point on the graph of a function to a certain straight line tends to zero as the point moves infinitely away from the origin, then the straight line is called asymptote of the graph of the function.

If there is a number such that , then the line is vertical asymptote.

If there are limits , then the line is oblique (horizontal at k=0) asymptote.

e) General study of function

1. Function domain

2. Points of intersection of the graph with the coordinate axes

3. Study of a function for continuity, even/odd and periodicity

4. Intervals of monotonicity of a function

5. Extremum points of the function

6. Convexity intervals and inflection points of a function graph

7. Asymptotes of the graph of a function

8. Function graph.

Task 5. Explore the function and construct its graph.

Solution. 1) The function is defined on the entire number line except for the point where the denominator of the fraction goes to zero. . We have: does not belong to the domain of definition of this function. Consequently, the stationary points of this function are the points with the minimum value (as shown in the figure).

8) Using the data obtained, let’s build a graph of the original function: