Whether the function is continuous or discontinuous. How to investigate a function for continuity
Determining the break point of a function
end point x 0
called function break point f (x), if the function is defined on some punctured neighborhood of the point x 0
, but is not continuous at this point.
That is, at the point of discontinuity, the function is either not defined or defined, but at least one one-sided limit at this point either does not exist, or is not equal to the value f (x0) functions at point x 0 . See "Determining the continuity of a function at a point".
Determination of the break point of the 1st kind
The point is called breaking point of the first kind, if is a breakpoint and there are finite one-sided limits on the left and on the right:
.
Function jump definition
Jump Δ function at a point is called the difference between the limits on the right and on the left
.
Determining a break point
The point is called break point if there is a limit
,
but the function at the point is either not defined or is not equal to the limit value: .
Thus, a discontinuity point is a discontinuity point of the first kind, at which the jump of the function zero.
Determination of the break point of the 2nd kind
The breaking point is called breaking point of the second kind, if it is not a discontinuity point of the 1st kind. That is, if there is not at least one one-sided limit, or at least one one-sided limit at a point is equal to infinity.
Investigation of functions for continuity
When investigating functions for continuity, we use the following facts.
- Elementary Functions and their inverses are continuous on their domain of definition. These include the following features:
, as well as the constant and their inverse functions. See "Elementary Functions Reference". - Sum, difference and product continuous, on some set of functions, is a continuous, function on this set.
Private of two continuous, on some set of functions, is a continuous, function on this set, except for the points at which the denominator of the fraction vanishes. See "Arithmetic Properties of Continuous Functions" - Complex function is continuous at a point if the function is continuous at a point and the function is continuous at a point . See "Limit and continuity of a complex function"
Examples
Example 1
Given a function and two argument values and . It is required: 1) to establish whether the given function is continuous or discontinuous for each of the given values of the argument; 2) in case of a function break, find its limits at the break point on the left and right, set the type of break; 3) make a schematic drawing.
.
The given function is complex. It can be viewed as a composition of two functions:
, . Then
.
Let's consider a function. It is composed of a function and constants using the arithmetic operations of addition and division. The function is an elementary - power function with an exponent 1
. It is defined and continuous for all values of the variable . Therefore, the function is defined and continuous for all , except for the points at which the denominator of the fraction vanishes. We equate the denominator to zero and solve the equation:
.
We get a single root.
So, the function is defined and continuous for all except the point .
Let's consider a function. This is an exponential function with a positive base. It is defined and continuous for all values of the variable .
So given function is defined and continuous for all values of the variable , except for the point .
Thus, at the point , the given function is continuous.
Graph of the function y = 4 1/(x+2).
Let's consider a point. At this point, the function is not defined. Therefore, it is not continuous. Let us establish the type of discontinuity. To do this, we find one-sided limits.
Using the connection between infinitely large and infinitely small functions, for the limit on the left we have:
at ,
,
,
.
Here we have used the following conventional notation:
.
We also used the property of the exponential function with base:
.
Similarly, for the limit on the right we have:
at ,
,
,
.
Since one of the one-sided limits is equal to infinity, there is a discontinuity of the second kind at the point.
At a point, the function is continuous.
At the point of discontinuity of the second kind,
.
Example 2
The function is set. Find function breakpoints, if they exist. Indicate the type of break and jumps of the function, if any. Make a drawing.
.
Graph of a given function.
The function is power function with an integer exponent equal to 1 . Such a function is also called a linear function. It is defined and continuous for all values of the variable .
There are two more functions included: and . They are composed of a function and constants using the arithmetic operations of addition and multiplication:
,
.
Therefore they are also continuous for all .
Since the functions included in the composition are continuous for all , it can have discontinuity points only at the gluing points of its components. These are points and . We investigate for continuity at these points. To do this, we find one-sided limits.
Let's consider a point. To find the left limit of a function at this point, we must use the values of this function in any left punctured neighborhood of the point . Let's take the neighborhood. On her. Then the limit on the left:
.
Here we have used the fact that the function is continuous at a point (as it is at any other point). Therefore, its left (and right) limit is equal to the value of the function at this point.
Find the right limit at the point . To do this, we must use the values of the function in any right punctured neighborhood of this point. Let's take the neighborhood. On her. Then the limit on the right:
.
Here we have also used the continuity of the function .
Since, at the point , the limit on the left is not equal to the limit on the right, then the function in it is not continuous - this is the break point. Since the one-sided limits are finite, this is a discontinuity point of the first kind. Jump function:
.
Now consider a point. In the same way, we calculate one-sided limits:
;
.
Since the function is defined at a point and the left limit is equal to the right one, the function is continuous at this point.
The function has a discontinuity of the first kind at the point . Function jump in it: . At other points, the function is continuous.
Example 3
Determine the discontinuity points of the function and investigate the nature of these points if
.
Let us use the fact that the linear function is defined and continuous for all . The specified function is composed of a linear function and constants using the arithmetic operations of addition, subtraction, multiplication, and division:
.
Therefore, it is defined and continuous for all , except for the points at which the denominator of the fraction vanishes.
Let's find these points. We equate the denominator to zero and solve the quadratic equation:
;
;
;
.
Then
.
We use the formula:
.
With its help, we decompose the numerator into factors:
.
Then the given function will take the form:
(P1) .
It is defined and continuous for all except the points and . Therefore, the points and are the discontinuity points of the function.
Divide the numerator and denominator of the fraction in (P1) by:
(P2) .
We can do this operation if . Thus,
at .
That is, the functions and differ only at one point: it is defined at , but it is not defined at this point.
To determine the genus of breakpoints, we need to find the one-sided limits of the function at the points and . To calculate them, we will use the fact that if the values of the function are changed, or made undefined at a finite number of points, then this will not have any effect on the value or existence of the limit at an arbitrary point (see "Influence of the values of the function at a finite number of points on the value of the limit "). That is, the limits of the function at any points are equal to the limits of the function.
Let's consider a point. The denominator of the fraction in the function , at does not vanish. Therefore, it is defined and continuous for . This implies that there is a limit at and it is equal to the value of the function at this point:
.
Therefore, the point is a discontinuity point of the first kind.
Let's consider a point. Using the connection of infinitesimal and infinitely large functions , we have:
;
.
Since the limits are infinite, there is a discontinuity of the second kind at this point.
The function has a discontinuity point of the first kind at , and a discontinuity point of the second kind at .
References:
O.I. Demons. Lectures on mathematical analysis. Part 1. Moscow, 2004.
4.1. Basic theoretical information
Definition. Function y=f(x) is called continuous at a point X 0 , if this function is defined in some neighborhood of the point X 0 and if
that is, an infinitesimal increment of the argument in a neighborhood of the point X 0 corresponds to an infinitesimal increment of the function .
Definition. Function y=f(x) is continuous at the point X 0 , if it is defined in some neighborhood of this point and if the limit of the function as the independent variable tends to X to X 0 exists and is equal to the value of the function at x=x 0 , i.e
Definition. Let be X→ X 0 , remaining all the time to the left of X 0 . If under this condition f(x) tends to the limit, then it is called the left limit of the function f(x) at the point X 0 , i.e
The right limit is defined similarly
Definition. The function is continuous at a point X 0 if:
function defined at point X 0 ;
there are left and right limits of the function f(x) at X→ X 0 ;
all three numbers (X 0 ), f(x 0 –0), f(x 0 +0) match, that is,
Definition. A function is called continuous on an interval if it is continuous at every point in it.
Theorem . If two functions f(x) and g(x) are defined in the same
interval and both are continuous at a point X 0 , then at the same point the functions
Theorem. A complex function consisting of a finite number of continuous functions is continuous.
All major elementary functions are continuous in their domain of definition .
Definition. If at any point X 0 function is not continuous, then the point X 0 is called the discontinuity point of the function, and the function itself is discontinuous at this point .
Definition. If at the point X 0 there is a finite lim f(x)= A
(the left-hand and right-hand limits exist, are finite and equal to each other), but it does not coincide with the value of the function at the point, or the function at the point is not defined, then the point X 0 called the break point . Received point image removable gap is shown in fig. one .
Definition. A discontinuity point of the first kind or a final discontinuity point is such a point X 0 , in which the function has left and right finite limits, but they are not equal to each other.
On fig. 2 shows a graphical representation of the discontinuity of a function of the first kind at a point X 0
Definition. If at least one of the limits f(x 0 – 0) or f(x 0 + 0) does not exist or is infinite, then the point X 0 is called a discontinuity point, of the second kind.
Graphical representations of discontinuities of functions of the second kind at a point X 0 shown in fig. 3 (a B C).
The above definitions of the continuity of a function f(x) at the point X 0
Example Investigate for continuity, determine the nature of break points,
represent in the neighborhood of discontinuity points the function
This is a rational function It is defined and continuous for all values X, Besides X= 1, since x = 1 denominator goes to zero . AT point x = 1 function is broken. Let us calculate the limit of this function for
X→ 1, we have
The final limit of the function at X→ 1 exists, and the function at the point
x = 1 not defined; means point X= 1 is a retractable discontinuity point.
If we redefine the function, that is, put f (1) = 5, then the function
will be continuous.
x = 1 is shown in fig.
4.
Comment. This function
undefined at x = 1, same as continuous function
at all points except X=1
Investigate the continuity of a function and determine the nature of its discontinuity points
The scope of the function is the entire numerical axis. On the intervals (–, 0), (0,+) the function is continuous . The gap is only possible at the point X= 0, in which the analytical definition of the function changes.
Let's find one-sided limits of the function :
The left and right limits, although finite, are not equal to each other. Therefore, at the point X= 0 the function has a discontinuity of the first kind. The jump of the function at the discontinuity point is
Behavior of a function in a neighborhood of a point x = 0 is shown in fig. 5.
Rice. 5
Example Explore Function f(x) on continuity, determine the nature of its discontinuity points, depict its behavior in the vicinity of discontinuity points.
The function is defined and continuous on the entire number axis, except for points X, = –2 and X 2 = 2, and
does not exist .
Calculate one-sided limits at a point X, = –2.
So at the point x = - 2 function suffers a discontinuity of the second kind. We study the nature of the discontinuity of the function at the point X 2 = 2. We have
At the point X 2 = 2 the function also suffers a discontinuity of the second kind.
Behavior of a function in a neighborhood of points X X = – 2 and X 2 = 2 shown in fig. 6 .
Explore Function f(x) = e x + i for continuity , determine the nature of the discontinuity points, depict the behavior of the function in the vicinity of the discontinuity points.
Function 
undefined at X= –3, so the function 
continuous for all 
Besides X= -3. Let us determine the nature of the discontinuity of the function. We have
that is, one of the limits is equal to infinity, which means that the function breaks
second kind .
Function Behavior f(x) = e x +3 in the vicinity of the discontinuity point x =-3 is shown in fig. 7
4.2. Exercises for independent work of students
1. Investigate functions for continuity, determine the nature of their discontinuity points, graphically depict the behavior of functions in the neighborhood
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2. Investigate functions for continuity, determine the nature of their discontinuity points, graphically depict the behavior of functions in the vicinity of discontinuity points
Definition. Let the function f(x) be defined on some interval and x 0 be a point of this interval. If , then f(x) is called continuous at the point x 0 .It follows from the definition that one can speak of continuity only with respect to those points at which f(x) is defined (no such condition was set when defining the limit of a function). For continuous functions
, that is, the operations f and lim commute. According to the two definitions of the limit of a function at a point, two definitions of continuity can be given - “in the language of sequences” and “in the language of inequalities” (in the language of ε-δ). It is suggested that you do it yourself. For practical use, it is sometimes more convenient to define continuity in terms of increments.
The value Δx=x-x 0 is called the increment of the argument, and Δy=f(x)-f(x 0) is the increment of the function when moving from point x 0 to point x.
Definition. Let f(x) be defined at the point x 0 . The function f(x) is called continuous at the point x 0 if an infinitesimal increment of the argument at this point corresponds to an infinitesimal increment of the function, that is, Δy→0 as Δx→0.
Example 1
Prove that the function y=sinx is continuous for any value of x.
Decision.
Let x 0 be an arbitrary point. Giving it an increment Δx, we get the point x=x 0 +Δx. Then 
. We get 
.
Definition.
The function y=f(x) is called continuous at the point x 0 on the right (left) if
.
A function continuous at an interior point will be both right and left continuous. The converse is also true: if a function is continuous at a point on the left and right, then it will be continuous at that point. However, the function can only be continuous on one side. For example, for 
, 
, f(1)=1, therefore, this function is continuous only on the left (for the graph of this function, see Section 5.7.2 above).
Definition.
A function is called continuous on some interval if it is continuous at every point of this interval.
In particular, if the interval is a segment , then one-sided continuity is implied at its ends.
Properties of continuous functions
1. All elementary functions are continuous in their domain of definition.2. If f(x) and φ(x), given on some interval, are continuous at the point x 0 of this interval, then the functions will also be continuous at this point.
3. If y=f(x) is continuous at a point x 0 from X, and z=φ(y) is continuous at the corresponding point y 0 =f(x 0) from Y, then the complex function z=φ(f(x )) will be continuous at the point x 0 .
Function breaks and their classification
A sign of the continuity of the function f (x) at the point x 0 is the equality, which implies the presence of three conditions:1) f(x) is defined at the point x 0 ;
2)
;
3) .
If at least one of these requirements is violated, then x 0 is called the break point of the function. In other words, a discontinuity point is a point where this function is not continuous. From the definition of breakpoints, it follows that the breakpoints of a function are:
a) points belonging to the domain of the function, at which f(x) loses the continuity property,
b) points that do not belong to the domain of f(x), which are adjacent points of two intervals of the domain of the function.
For example, for a function, the point x=0 is a break point, since the function at this point is not defined, and the function
has a discontinuity at the point x=1, which is adjacent for two intervals (-∞,1) and (1,∞) of the domain f(x) and does not exist. The following classification is accepted for discontinuity points.
1) If at the point x 0 there are finite 
and 
, but f(x 0 +0)≠f(x 0 -0), then x 0 is called breaking point of the first kind
, while they call jump function
.
Example 2
Consider the function 
The break of the function is possible only at the point x=2 (at other points it is continuous like any polynomial). 
Let's find 
, 
. Since the one-sided limits are finite, but not equal to each other, at the point x=2 the function has a discontinuity of the first kind. notice, that 
, hence the function is right-continuous at this point (Fig. 2).
2) Discontinuity points of the second kind
points are called at which at least one of the one-sided limits is equal to ∞ or does not exist.
Example 3
The function y=2 1/ x is continuous for all values of x, except for x=0. Find one-sided limits: 
, 
, hence x=0 is a discontinuity point of the second kind (Fig. 3).
3) The point x=x 0 is called break point
, if f(x 0 +0)=f(x 0 -0)≠f(x 0).
The gap is “removable” in the sense that it is enough to change (redefine or redefine) the value of the function at this point by setting , and the function will become continuous at the point x 0 . 
Example 4
It is known that 
, and this limit does not depend on how x tends to zero. But the function at the point x=0 is not defined. If we extend the definition of the function by setting f(0)=1, then it turns out to be continuous at this point (at other points it is continuous as a quotient of the continuous functions sinx and x). 
Example 5
Investigate for continuity a function 
.
Decision.
The functions y=x 3 and y=2x are defined and continuous everywhere, including in the indicated intervals. Let's examine the junction point of the gaps x=0: 
, 
, . We get that , whence it follows that at the point x=0 the function is continuous.
Definition.
A function that is continuous on an interval except for a finite number of discontinuities of the first kind or a removable discontinuity is said to be piecewise continuous on this interval.
Examples of discontinuous functions
Example 1
The function is defined and continuous on (-∞,+∞) except for the point x=2. Let's define the type of break. Insofar as 
and 
, then at the point x=2 there is a discontinuity of the second kind (Fig. 6). 
Example 2
The function is defined and continuous for all x except x=0, where the denominator is zero. Let's find one-sided limits at the point x=0: The one-sided limits are finite and different, therefore, x=0 is a discontinuity point of the first kind (Fig. 7).
Example 3
Determine at what points and what kind of discontinuities the function has 
This function is defined on [-2,2]. Since x 2 and 1/x are continuous, respectively, in the intervals [-2,0] and , the gap can only be at the junction of the intervals, that is, at the point x=0. Since , then x=0 is a discontinuity point of the second kind.
Example 4
Is it possible to eliminate breaks in functions:
a) 
at the point x=2;
b) 
at the point x=2;
in) 
at the point x=1?
Decision.
About example a), we can immediately say that the discontinuity f(x) at the point x=2 cannot be eliminated, since there are infinite one-sided limits at this point (see example 1).
b) The function g(x) although has finite one-sided limits at the point x=2
(
,
),
but they do not match, so the gap cannot be closed either.
c) The function φ(x) at the discontinuity point x=1 has equal one-sided finite limits: . Therefore, the gap can be eliminated by redefining the function at the point x=1 by putting f(1)=1 instead of f(1)=2.
Example 5 Show that the Dirichlet function
discontinuous at every point on the numerical axis.
Decision. Let x 0 be any point from (-∞,+∞). In any of its neighborhoods, there are both rational and irrational points. This means that in any neighborhood x 0 the function will have values equal to 0 and 1. In this case, there cannot be a limit of the function at the point x 0 either on the left or on the right, which means that the Dirichlet function at each point of the real axis has discontinuities of the second kind.
Example 6 Find function break points
and determine their type.
Decision. Points suspected of breaking are points x 1 =2, x 2 =5, x 3 =3.
At the point x 1 =2 f(x) has a discontinuity of the second kind, since
.
The point x 2 =5 is a point of continuity, since the value of the function at this point and in its vicinity is determined by the second line, not the first: .
Let's explore the point x 3 =3: ,
, whence it follows that x=3 is a discontinuity point of the first kind. For independent decision.
Investigate functions for continuity and determine the type of discontinuity points:
1) 
; Answer: x=-1 – break point;
2) 
; Answer: Discontinuity of the second kind at the point x=8;
3) 
; Answer: Discontinuity of the first kind at x=1;
4) 
Answer: At the point x 1 \u003d -5 there is a removable gap, at x 2 \u003d 1 - a gap of the second kind and at the point x 3 \u003d 0 - a gap of the first kind.
5) How should the number A be chosen so that the function 
would be continuous at the point x=0?
Answer: A=2.
6) Is it possible to choose the number A so that the function 
would be continuous at the point x=2?
Answer: no.
Odd functions
An odd power where is an arbitrary integer.
Sinus.
· Tangent.
An even power where is an arbitrary integer.
· Cosine.
· Absolute value (modulus) .
Periodic function is a function that repeats its values at some regular interval of the argument, i.e., does not change its value when some fixed nonzero number is added to the argument ( period functions) over the entire domain of definition.
· More formally, a function is called periodic if there is such a number T≠0 (period) that the equality is true on the entire domain of the function.
· Based on the definition, for a periodic function, the equality is also true, where is any integer.
· All trigonometric functions are periodic.
3) Zeros (roots) of a function are the points where it vanishes.
Finding the point of intersection of the graph with the axis Oy. To do this, you need to calculate the value f(0). Find also the points of intersection of the graph with the axis Ox, why find the roots of the equation f(x) = 0 (or make sure there are no roots).
The points where the graph intersects the axis are called function zeros. To find the zeros of the function, you need to solve the equation, that is, find those x values, for which the function vanishes.
4) Intervals of constancy of signs, signs in them.
Intervals where the function f(x) retains its sign.
The constancy interval is the interval at every point in which function is positive or negative.
ABOVE the x-axis.
BELOW axis.
5) Continuity (points of discontinuity, character of discontinuity, asymptotes).
continuous function - a function without "jumps", that is, one in which small changes in the argument lead to small changes in the value of the function.
Removable breakpoints
If the limit of the function exist, but the function is not defined at this point, or the limit does not match the value of the function at this point:
then the point is called break point functions (in complex analysis, a removable singular point).
If we “correct” the function at the point of a removable discontinuity and put , then we get a function that is continuous at this point. Such an operation on a function is called extending the function to continuous or extension of the function by continuity, which justifies the name of the point, as points disposable gap.
Discontinuity points of the first and second kind
If the function has a discontinuity at a given point (that is, the limit of the function at a given point is absent or does not match the value of the function at a given point), then for numerical functions there are two possible options associated with the existence of numerical functions unilateral limits:
if both one-sided limits exist and are finite, then such a point is called breaking point of the first kind. Removable discontinuity points are discontinuity points of the first kind;
if at least one of the one-sided limits does not exist or is not a finite value, then such a point is called breaking point of the second kind.
Asymptote- a straight line with the property that the distance from a point of the curve to this straight line tends to zero when the point is removed along the branch to infinity.
vertical
The vertical asymptote is a straight line of the form under the condition that the limit exists.
As a rule, when determining the vertical asymptote, they look for not one limit, but two one-sided ones (left and right). This is done in order to determine how the function behaves as it approaches the vertical asymptote with different sides. For example:
Horizontal
Horizontal asymptote - a straight line of the form subject to the existence of a limit
oblique
Oblique asymptote - a straight line of the form subject to the existence of limits
Note: A function can have no more than two oblique (horizontal) asymptotes.
Note: if at least one of the two limits mentioned above does not exist (or is equal to ), then the oblique asymptote at (or ) does not exist.
if in item 2.), then , and the limit is found by the horizontal asymptote formula, .
6) Finding intervals of monotonicity. Find monotonicity intervals of a function f(x) (that is, intervals of increase and decrease). This is done by examining the sign of the derivative f (x). To do this, find the derivative f (x) and solve the inequality f (x) 0. On the intervals where this inequality is satisfied, the function f(x) increases. Where the reverse inequality holds f (x) 0, function f(x) decreases.
Finding a local extremum. Having found the intervals of monotonicity, we can immediately determine the points of a local extremum where the increase is replaced by a decrease, there are local maxima, and where the decrease is replaced by an increase, local minima. Calculate the value of the function at these points. If a function has critical points that are not local extremum points, then it is useful to calculate the value of the function at these points as well.
Finding the largest and smallest values of the function y = f(x) on a segment(continuation)
| 1.Find the derivative of the function: f (x). 2. Find the points where the derivative is equal to zero: f (x)=0 x 1, x 2 ,... 3. Determine the belonging of points X 1 ,X 2 , … segment [ a; b]: let be x 1 a;b, a x 2 a;b. 4. Find the values of the function at the selected points and at the ends of the segment: f(x 1), f(x 2),..., f(x a),f(xb), 5. Selecting the largest and smallest values of the function from those found. Comment. If on the segment [ a; b] there are discontinuity points, then it is necessary to calculate one-sided limits in them, and then take their values into account in choosing the largest and smallest values of the function. |
7) Finding intervals of convexity and concavity. This is done by examining the sign of the second derivative f (x). Find the inflection points at the junctions of the convex and concavity intervals. Calculate the value of the function at the inflection points. If the function has other points of continuity (other than inflection points) at which the second derivative is equal to 0 or does not exist, then at these points it is also useful to calculate the value of the function. Finding f (x) , we solve the inequality f (x) 0. On each of the intervals of the solution, the function will be downward convex. Solving the reverse inequality f (x) 0, we find the intervals where the function is convex upwards (that is, concave). We define inflection points as those points at which the function changes the direction of convexity (and is continuous).
The point a is called the point of a removable discontinuity of the function, if the limit of the function at this point exists, but at the point a the function is either not defined, or its value is not equal to the limit at this point
Discontinuity of the first kind.
Point a is called a discontinuity point of the first kind of a function if at this point the function has finite, but not equal, left and right limits.
Discontinuity of the second kind.
A point a is called a discontinuity point of the second kind of a function A point a is called a discontinuity point of a function if at this point the function does not have at least one of the one-sided limits or at least one of the one-sided limits is infinite.
25. Derivative: definition, mechanical and geometric meaning. The equation of the tangent to the curve.
Derivative Definition
Let the function is defined on some interval X. Let us give the value of the argument at the point an arbitrary increment so that the point also belonged to X. Then the corresponding increment functions will amount to .
ODA. The derivative of a function at a point is the limit of the ratio of the increment of the function at this point to the increment of the argument at (if this limit exists).
If the limit is infinite at some point, then the function is said to have an infinite derivative at that point. If a function has a derivative at every point in X, then the derivative is also a function of x defined on X.
The geometric meaning of the derivative
To clarify the geometric meaning of the derivative, we need the definition of a tangent to the graph of a function at a given point.
Def. The tangent to the graph of the function at the point M is the limiting position of the secant MN when the point N tends to the point M along the curve.
The equation of a pencil of lines passing through a point has the form
The slope of the secant is equal to
Then the slope of the tangent is
From this follows a clear conclusion that . This is what it consists geometric meaning of the derivative.
from here v(t 0)= x'(t 0) , i.e. Velocity is the derivative of the coordinate with respect to time. This is what it consists mechanical sense derivative . Likewise, acceleration is the derivative of speed with respect to time: a = v'(t).
Tangent equation to the graph of a function at a point has the form:
26. Basic rules of differentiation. Derivatives of basic elementary functions.
Differentiation rules.
1. The derivative of a constant is zero
2. The derivative of the argument is equal to one.
3. The derivative of the algebraic sum of a finite number of differentiable functions is equal to the same sum of the derivatives of these functions.
The derivative of the product of two differentiable functions is equal to the product of the derivative of the first factor by the second plus the product of the first factor by the derivative of the second.
Corollary 1. The constant factor can be taken out of the sign of the derivative.
Corollary 2. The derivative of the product of several differentiable functions is equal to the sum of the products of the derivative of each of the factors and all the others, for example
5. The derivative of the quotient of two differentiable functions can be found by the formula:
Derivatives of basic elementary functions.
1. (C)” = 0, where C = const
2. (x a)” = ax a-1 , where a is not equal to 0
3. (a x)” = a x ln a, where a > 0
4. (e x)” = e x
5. (log a x)” =1/x ln a , where a > 0
6. (ln x)” =1/x
7. (sin x)" = cos x
8. (cos x)” = - sin x
9. (tan x)” =1/cos 2 x
10. (ctg x)” = -1/sin 2 x
11. (arcsin x)" = 1/~1-x 2
12. (arccos x)' = -1/~1-x 2
13. (arctg x)” =1/1+x 2
14. (arcctg x)” = -1/1+x 2
27. Derivative of a complex function. Derivatives of higher orders.
