Derivative by definition (via a limit). Examples of solutions. Posts tagged "derivative definition"

When deciding various tasks geometry, mechanics, physics and other branches of knowledge became necessary using the same analytical process from this function y=f(x) receive new feature which is called derivative function(or just derivative) of a given function f(x) and is designated by the symbol

The process by which from a given function f(x) get a new feature f" (x), called differentiation and it consists of the following three steps: 1) give the argument x increment  x and determine the corresponding increment of the function  y = f(x+ x) -f(x); 2) make up a relation

3) counting x constant and  x0, we find
, which we denote by f" (x), as if emphasizing that the resulting function depends only on the value x, at which we go to the limit. Definition: Derivative y " =f " (x) given function y=f(x) for a given x is called the limit of the ratio of the increment of a function to the increment of the argument, provided that the increment of the argument tends to zero, if, of course, this limit exists, i.e. finite. Thus,
, or

Note that if for some value x, for example when x=a, attitude
at  x0 does not tend to the finite limit, then in this case they say that the function f(x) at x=a(or at the point x=a) has no derivative or is not differentiable at the point x=a.

2. Geometric meaning of the derivative.

Consider the graph of the function y = f (x), differentiable in the vicinity of the point x 0

f(x)

Let's consider an arbitrary straight line passing through a point on the graph of a function - point A(x 0, f (x 0)) and intersecting the graph at some point B(x;f(x)). Such a line (AB) is called a secant. From ∆ABC: AC = ∆x; BC =∆у; tgβ=∆y/∆x.

Since AC || Ox, then ALO = BAC = β (as corresponding for parallel). But ALO is the angle of inclination of the secant AB to the positive direction of the Ox axis. This means that tanβ = k is the slope of straight line AB.

Now we will reduce ∆x, i.e. ∆х→ 0. In this case, point B will approach point A according to the graph, and secant AB will rotate. The limiting position of the secant AB at ∆x→ 0 will be a straight line (a), called the tangent to the graph of the function y = f (x) at point A.

If we go to the limit as ∆x → 0 in the equality tgβ =∆y/∆x, we get
ortg =f "(x 0), since
-angle of inclination of the tangent to the positive direction of the Ox axis
, by definition of a derivative. But tg = k is the angular coefficient of the tangent, which means k = tg = f "(x 0).

So, geometric meaning the derivative is as follows:

Derivative of a function at point x 0 equal to the slope of the tangent to the graph of the function drawn at the point with the abscissa x 0 .

3. Physical meaning of the derivative.

Consider the movement of a point along a straight line. Let the coordinate of a point at any time x(t) be given. It is known (from a physics course) that the average speed over a period of time is equal to the ratio of the distance traveled during this period of time to the time, i.e.

Vav = ∆x/∆t. Let's go to the limit in the last equality as ∆t → 0.

lim Vav (t) = (t 0) - instantaneous speed at time t 0, ∆t → 0.

and lim = ∆x/∆t = x"(t 0) (by definition of derivative).

So, (t) =x"(t).

The physical meaning of the derivative is as follows: derivative of the functiony = f(x) at pointx 0 is the rate of change of the functionf(x) at pointx 0

The derivative is used in physics to find velocity from a known function of coordinates versus time, acceleration from a known function of velocity versus time.

(t) = x"(t) - speed,

a(f) = "(t) - acceleration, or

If the law of motion of a material point in a circle is known, then one can find the angular velocity and angular acceleration during rotational motion:

φ = φ(t) - change in angle over time,

ω = φ"(t) - angular velocity,

ε = φ"(t) - angular acceleration, or ε = φ"(t).

If the law of mass distribution of an inhomogeneous rod is known, then the linear density of the inhomogeneous rod can be found:

m = m(x) - mass,

x  , l - length of the rod,

p = m"(x) - linear density.

Using the derivative, problems from the theory of elasticity and harmonic vibrations are solved. So, according to Hooke's law

F = -kx, x – variable coordinate, k – spring elasticity coefficient. Putting ω 2 =k/m, we obtain the differential equation of the spring pendulum x"(t) + ω 2 x(t) = 0,

where ω = √k/√m oscillation frequency (l/c), k - spring stiffness (H/m).

An equation of the form y" + ω 2 y = 0 is called the equation of harmonic oscillations (mechanical, electrical, electromagnetic). The solution to such equations is the function

y = Asin(ωt + φ 0) or y = Acos(ωt + φ 0), where

A - amplitude of oscillations, ω - cyclic frequency,

φ 0 - initial phase.

Create a ratio and calculate the limit.

Where did it come from? table of derivatives and differentiation rules? Thanks to the only limit. It seems like magic, but in reality it is sleight of hand and no fraud. In class What is a derivative? I started looking at specific examples, where, using the definition, I found the derivatives of linear and quadratic function. For the purpose of cognitive warm-up, we will continue to disturb table of derivatives, honing the algorithm and technical solutions:

Example 1

Essentially, you need to prove special case derivative power function, which usually appears in the table: .

Solution technically formalized in two ways. Let's start with the first, already familiar approach: the ladder starts with a plank, and the derivative function starts with the derivative at a point.

Let's consider some(specific) point belonging to domain of definition function in which there is a derivative. Let us set the increment at this point (of course, not going beyondo/o -I) and compose the corresponding increment of the function:

Let's calculate the limit:

The uncertainty 0:0 is eliminated by a standard technique, considered back in the first century BC. Multiply the numerator and denominator by the conjugate expression :

The technique for solving such a limit is discussed in detail at introductory lesson about the limits of functions.

Since you can choose ANY point of the interval as quality, then, having made the replacement, we get:

Answer

Once again let's rejoice at logarithms:

Example 2

Find the derivative of a function using the definition of derivative

Solution: Let's consider a different approach to promoting the same task. It is exactly the same, but more rational in terms of design. The idea is to get rid of the subscript at the beginning of the solution and use the letter instead of the letter.

Let's consider arbitrary point belonging to domain of definition function (interval) and set the increment in it. But here, by the way, as in most cases, you can do without any reservations, since the logarithmic function is differentiable at any point in the domain of definition.

Then the corresponding increment of the function is:

Let's find the derivative:

The simplicity of the design is balanced by the confusion that may arise for beginners (and not only). After all, we are accustomed to the fact that the letter “X” changes in the limit! But here everything is different: - an antique statue, and - a living visitor, briskly walking along the museum corridor. That is, “x” is “like a constant.”

I will comment on the elimination of uncertainty step by step:

(1) We use the property of the logarithm .

(2) In parentheses, divide the numerator by the denominator term by term.

(3) In the denominator, we artificially multiply and divide by “x” to take advantage of remarkable limit , while as infinitesimal stands out.

Answer: by definition of derivative:

Or in short:

I propose to construct two more table formulas yourself:

Example 3

In this case, it is convenient to immediately reduce the compiled increment to a common denominator. An approximate sample of the assignment at the end of the lesson (first method).

Example 3:Solution : consider some point , belonging to the domain of definition of the function . Let us set the increment at this point and compose the corresponding increment of the function:

Let's find the derivative at the point :

Since as a you can select any point function domain , That And
Answer : by definition of derivative

Example 4

Find derivative by definition

And here everything needs to be reduced to wonderful limit. The solution is formalized in the second way.

A number of other tabular derivatives. Full list can be found in a school textbook, or, for example, the 1st volume of Fichtenholtz. I don’t see much point in copying proofs of differentiation rules from books - they are also generated by the formula.

Example 4:Solution , belonging to , and set the increment in it

Let's find the derivative:

Using a wonderful limit

Answer : by definition

Example 5

Find the derivative of a function , using the definition of derivative

Solution: we use the first design style. Let's consider some point belonging to , and specify the increment of the argument at it. Then the corresponding increment of the function is:

Perhaps some readers have not yet fully understood the principle by which increments need to be made. Take a point (number) and find the value of the function in it: , that is, into the function instead of"X" should be substituted. Now we also take a very specific number and also substitute it into the function instead of"iksa": . We write down the difference, and it is necessary completely put in brackets.

Compiled function increment It can be beneficial to immediately simplify. For what? Facilitate and shorten the solution to a further limit.

We use formulas, open the brackets and reduce everything that can be reduced:

The turkey is gutted, no problem with the roast:

As a result:

Since you can choose any quality real number, then we make the replacement and get .

Answer: by definition.

For verification purposes, let’s find the derivative using differentiation rules and tables:

It is always useful and pleasant to know the correct answer in advance, so it is better to differentiate the proposed function in a “quick” way, either mentally or in a draft, at the very beginning of the solution.

Example 6

Find the derivative of a function by definition of derivative

This is an example for independent decision. The result is obvious:

Example 6:Solution : consider some point , belonging to , and set the increment of the argument in it . Then the corresponding increment of the function is:

Let's calculate the derivative:

Thus:
Because as you can choose any real number, then And
Answer : by definition.

Let's go back to style #2:

Example 7

Let's find out immediately what should happen. By differentiation rule complex function :

Solution: consider an arbitrary point belonging to , set the increment of the argument at it and compose the increment of the function:

Let's find the derivative:

(1) Use trigonometric formula .

(2) Under the sine we open the brackets, under the cosine we present similar terms.

(3) Under the sine we reduce the terms, under the cosine we divide the numerator by the denominator term by term.

(4) Due to the oddness of the sine, we take out the “minus”. Under the cosine we indicate that the term .

(5) We carry out artificial multiplication in the denominator in order to use first wonderful limit. Thus, the uncertainty is eliminated, let’s tidy up the result.

Answer: by definition

As you can see, the main difficulty of the problem under consideration rests on the complexity of the limit itself + a slight uniqueness of the packaging. In practice, both methods of design occur, so I describe both approaches in as much detail as possible. They are equivalent, but still, in my subjective impression, it is more advisable for dummies to stick to option 1 with “X-zero”.

Example 8

Using the definition, find the derivative of the function

Example 8:Solution : consider an arbitrary point , belonging to , let us set the increment in it and compose the increment of the function:

Let's find the derivative:

We use the trigonometric formula and the first remarkable limit:

Answer : by definition

Let's look at a rarer version of the problem:

Example 9

Find the derivative of the function at the point using the definition of derivative.

Firstly, what should be the bottom line? Number

Let's calculate the answer in the standard way:

Solution: from the point of view of clarity, this task is much simpler, since the formula instead considers a specific value.

Let's set the increment at the point and compose the corresponding increment of the function:

Let's calculate the derivative at the point:

We use a very rare tangent difference formula and once again we reduce the solution to the first wonderful limit:

Answer: by definition of derivative at a point.

The problem is not so difficult to solve and “in general view" - it is enough to replace with or simply depending on the design method. In this case, it is clear that the result will not be a number, but a derived function.

Example 10

Using the definition, find the derivative of the function at a point (one of which may turn out to be infinite), which I’m talking about general outline already told on theoretical lesson about derivative.

Some piecewise defined functions are also differentiable at the “junction” points of the graph, for example, catdog has a common derivative and a common tangent (x-axis) at the point. Curve, but differentiable by ! Those interested can verify this for themselves using the example just solved.

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Solving physical problems or examples in mathematics is completely impossible without knowledge of the derivative and methods for calculating it. Derivative is one of the most important concepts mathematical analysis. We decided to devote today’s article to this fundamental topic. What is a derivative, what is its physical and geometric meaning, how to calculate the derivative of a function? All these questions can be combined into one: how to understand the derivative?

Geometric and physical meaning of derivative

Let there be a function f(x) , specified in a certain interval (a, b) . Points x and x0 belong to this interval. When x changes, the function itself changes. Changing the argument - the difference in its values x-x0 . This difference is written as delta x and is called argument increment. A change or increment of a function is the difference between the values of a function at two points. Definition of derivative:

The derivative of a function at a point is the limit of the ratio of the increment of the function at a given point to the increment of the argument when the latter tends to zero.

Otherwise it can be written like this:

What's the point of finding such a limit? And here's what it is:

the derivative of a function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at a given point.

Physical meaning derivative: the derivative of the path with respect to time is equal to the speed of rectilinear motion.

Indeed, since school days everyone knows that speed is a particular path x=f(t) and time t . Average speed over a certain period of time:

To find out the speed of movement at a moment in time t0 you need to calculate the limit:

Rule one: set a constant

The constant can be taken out of the derivative sign. Moreover, this must be done. When solving examples in mathematics, take it as a rule - If you can simplify an expression, be sure to simplify it .

Example. Let's calculate the derivative:

Rule two: derivative of the sum of functions

The derivative of the sum of two functions is equal to the sum of the derivatives of these functions. The same is true for the derivative of the difference of functions.

We will not give a proof of this theorem, but rather consider a practical example.

Find the derivative of the function:

Rule three: derivative of the product of functions

The derivative of the product of two differentiable functions is calculated by the formula:

Example: find the derivative of a function:

Solution:

It is important to talk about calculating derivatives of complex functions here. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable.

In the above example we come across the expression:

In this case, the intermediate argument is 8x to the fifth power. In order to calculate the derivative of such an expression, we first calculate the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the intermediate argument itself with respect to the independent variable.

Rule four: derivative of the quotient of two functions

Formula for determining the derivative of the quotient of two functions:

We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it seems, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.

With any questions on this and other topics, you can contact the student service. For short term We will help you solve the most difficult tests and solve problems, even if you have never done derivative calculations before.

Ministry of Education of the Russian Federation

“MATI” - RUSSIAN STATE

TECHNOLOGICAL UNIVERSITY named after. K. E. TSIOLKOVSKY

Department of “Higher Mathematics”

Course assignment options

Guidelines for the course assignment

“Limits of functions. Derivatives"

Kulakova R. D.

Titarenko V. I.

Moscow 1999

Annotation

The proposed guidelines are aimed at helping first-year students master theoretical and practical material on the topic “Mathematical Analysis”.

In each section, after the theoretical part, typical problems are analyzed.

The guidelines cover the following topics: limits of functions, differentiation of functions given in various forms, derivatives and differentials of higher orders, L'Hopital's rule, application of the derivative to problems of geometry and mechanics.

To consolidate the material, students are asked to complete coursework on the topics listed above.

These guidelines can be used in all faculties and specialties.

1. Function limits

Some well-known techniques are used to determine the limits of sequences and functions:

If you need to find a limit

can be preliminarily reduced to a common denominator

Dividing by the term that has the maximum degree, we get a constant value in the numerator, and all terms tending to 0 in the denominator, that is

Then substituting x=a, we get:
;

4.
, when substituting x=0, we get
.

5. However, if it is necessary to find the limit of a rational function

, then when dividing by the term with the minimum degree, we get

; and, directing x to 0, we get:

If the limits contain irrational expressions, then it is necessary to introduce new variables to obtain a rational expression, or to transfer irrationalities from the denominator to the numerator and vice versa.

6.
; Let's make a variable change. We will replace
, at
, we get
.

7.
. If the numerator and denominator are multiplied by the same number, the limit does not change. Multiply the numerator by
and divide by the same expression so that the limit does not change, and multiply the denominator by
and divide by the same expression. Then we get:

The following remarkable limits are often used to define limits:

; (1)

. (2)

8.
.

To calculate such a limit, we reduce it to the 1st remarkable limit (1). To do this, multiply and divide the numerator by
, and the denominator is
, Then.

9.
To calculate this limit, we reduce it to the second remarkable limit. For this purpose, we select the whole part from the rational expression in brackets and present it in the form proper fraction. This is done in cases where
, Where
, A
, Where
;

, A
, then finally
. Here the continuity of the composition of continuous functions was used.

2. Derivative

Derivative of a function
called final limit the relationship between the increment of a function and the increment of the argument when the latter tends to zero:

, or
.

Geometrically, the derivative is the slope of the tangent to the graph of the function
at point x, that is
.

The derivative is the rate of change of a function at point x.

Finding the derivative is called differentiating the function.

Formulas for differentiating basic functions:

3. Basic rules of differentiation

Let then:

7) If , that is
, Where
And
have derivatives, then
(rule for differentiating a complex function).

4. Logarithmic differentiation

If you need to find from Eq.
, then you can:

a) logarithm both sides of the equation

b) differentiate both sides of the resulting equality, where
there is a complex function of x,

c) replace its expression in terms of x

Example:

5. Differentiation of implicit functions

Let the equation
defines as an implicit function of x.

a) differentiate both sides of the equation with respect to x
, we obtain an equation of the first degree with respect to ;

b) from the resulting equation we express .

Example:
.

6. Differentiation of functions given

parametrically

Let the function be given by parametric equations
,

Then
, or

Example:

7. Application of the derivative to problems

geometry and mechanics

Let
And
, Where - the angle formed with the positive direction of the OX axis by the tangent to the curve at the abscissa point .

Equation of a tangent to a curve
at the point
has the form:

, Where -derivative at
.

The normal to a curve is a line perpendicular to the tangent and passing through the point of tangency.

The normal equation has the form

Angle between two curves
And
at the point of their intersection
is the angle between the tangents to these curves at a point
. This angle is found by the formula

8. Higher order derivatives

If is the derivative of the function
, then the derivative of is called the second derivative, or derivative of the second order and is denoted , or
, or .

Derivatives of any order are defined similarly: third order derivative
; nth order derivative:

For the product of two functions, you can obtain a derivative of any nth order using the Leibniz formula:

9. Second derivative of an implicit function

-the equation determines , as an implicit function of x.

a) define
;

b) differentiate with respect to x the left and right sides of the equality
,

Moreover, differentiating the function
by variable x, remember that there is a function of x:

;

c) replacing through
, we get:
etc.

10. Derivatives of functions specified parametrically

Find
If
.

11. Differentials of the first and higher orders

First order differential of the function
is called the main part, linear with respect to the argument. The differential of an argument is the increment of an argument:
.

The differential of a function is equal to the product of its derivative and the differential of the argument:

Basic properties of the differential:

Where
.

If the increment
the argument is not enough absolute value, That
And.

Thus, the differential of a function can be used for approximate calculations.

Second order differential of the function
is called the differential of the first order differential:
.

Likewise:
.

If
And is an independent variable, then differentials of higher orders are calculated using the formulas

Find the first and second order differentials of the function

12. Calculation of limits using L'Hopital's rule

All of the above limits did not use the apparatus of differential calculus. However, if you need to find

and at
both of these functions are infinitesimal or both are infinitely large, then their ratio is not defined at the point
and therefore represents an uncertainty type or respectively. Because this is a relationship at a point
may have a limit, finite or infinite, then finding this limit is called the disclosure of uncertainty (L'Hopital Bernoulli's rule),

and the following equality holds:

, If
And
.

=
.

A similar rule holds if
And
, i.e.
.

=
.

L'Hopital's rule also makes it possible to resolve uncertainties of the type
And
. To calculate
, Where
- infinitesimal, and
- infinitely large at
(type uncertainty disclosure
) the product should be converted to the form

(uncertainty of type) or to species (type uncertainty ) and then use Lapital's rule.

To calculate
, Where
And
- infinitely large at
(type uncertainty disclosure
) the difference should be converted to the form
, then reveal the uncertainty type . If
, That
.

If
, then we get an uncertainty of the type (
), which is revealed similarly to example 12).

Because
, then we end up with an uncertainty of the type
and then we have

L'Hopital's rule can also be used to resolve uncertainties of the type
. In these cases, we mean calculating the limit of the expression
, Where
in case
is infinitesimal, in the case
- infinitely large, and in the case
- a function whose limit is equal to unity.

Function
in the first two cases it is an infinitesimal, and in the last case it is an infinitely large function.

Before looking for the limit of such expressions, they are taken logarithmically, i.e. If
, That
, then find the limit
, and then find the limit . In all of the above cases
is a type uncertainty
, which is opened similarly to example 12).

(use L'Hopital's rule)=

=
.

In this product of limits, the first factor is equal to 1, the second factor is the first remarkable limit and it is also equal to 1, and the last factor tends to 0, therefore:

and then
.