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Derivative of a function. Comprehensive Guide (2019)

Let's imagine a straight road passing through a hilly area. That is, it goes up and down, but does not turn right or left. If the axis is directed horizontally along the road and vertically, then the road line will be very similar to the graph of some continuous function:

The axis is a certain level of zero altitude; in life we ​​use sea level as it.

As we move forward along such a road, we also move up or down. We can also say: when the argument changes (movement along the abscissa axis), the value of the function changes (movement along the ordinate axis). Now let's think about how to determine the “steepness” of our road? What kind of value could this be? It’s very simple: how much the height will change when moving forward a certain distance. After all, on different areas roads, moving forward (along the x-axis) by one kilometer, we will rise or fall by different quantities meters relative to sea level (along the ordinate axis).

Let’s denote progress (read “delta x”).

The Greek letter (delta) is commonly used in mathematics as a prefix meaning "change". That is - this is a change in quantity, - a change; then what is it? That's right, a change in magnitude.

Important: an expression is a single whole, one variable. Never separate the “delta” from the “x” or any other letter!

That is, for example, .

So, we have moved forward, horizontally, by. If we compare the line of the road with the graph of a function, then how do we denote the rise? Certainly, . That is, as we move forward, we rise higher. The value is easy to calculate: if at the beginning we were at a height, and after moving we found ourselves at a height, then. If end point

turned out to be lower than the initial one, it will be negative - this means that we are not ascending, but descending.

Let's return to "steepness": this is a value that shows how much (steeply) the height increases when moving forward one unit of distance:

Now let's look at the top of a hill. If you take the beginning of the section half a kilometer before the summit, and the end half a kilometer after it, you can see that the height is almost the same.

That is, according to our logic, it turns out that the slope here is almost equal to zero, which is clearly not true. Just over a distance of kilometers a lot can change. Smaller areas need to be considered for more adequate and accurate assessment steepness. For example, if you measure the change in height as you move one meter, the result will be much more accurate. But even this accuracy may not be enough for us - after all, if there is a pole in the middle of the road, we can simply pass it. What distance should we choose then? Centimeter? Millimeter? Less is better!

IN real life Measuring distances to the nearest millimeter is more than enough. But mathematicians always strive for perfection. Therefore, the concept was invented infinitesimal, that is, the absolute value is less than any number that we can name. For example, you say: one trillionth! How much less? And you divide this number by - and it will be even less. And so on. If we want to write that a quantity is infinitesimal, we write like this: (we read “x tends to zero”). It is very important to understand that this number is not zero! But very close to it. This means that you can divide by it.

The concept opposite to infinitesimal is infinitely large (). You've probably already come across it when you were working on inequalities: this number is modulo greater than any number you can think of. If you come up with the biggest number possible, just multiply it by two and you'll get an even bigger number. And infinity still Furthermore what will happen. In fact, the infinitely large and the infinitely small are the inverse of each other, that is, at, and vice versa: at.

Now let's get back to our road. The ideally calculated slope is the slope calculated for an infinitesimal segment of the path, that is:

I note that with an infinitesimal displacement, the change in height will also be infinitesimal. But let me remind you that infinitesimal does not mean equal to zero. If you divide infinitesimal numbers by each other, you can get quite regular number, For example, . That is, one small value can be exactly times larger than another.

What is all this for? The road, the steepness... We’re not going on a car rally, but we’re teaching mathematics. And in mathematics everything is exactly the same, only called differently.

Concept of derivative

The derivative of a function is the ratio of the increment of the function to the increment of the argument for an infinitesimal increment of the argument.

Incrementally in mathematics they call change. The extent to which the argument () changes as it moves along the axis is called argument increment and is designated. How much the function (height) has changed when moving forward along the axis by a distance is called function increment and is designated.

So, the derivative of a function is the ratio to when. We denote the derivative with the same letter as the function, only with a prime on the top right: or simply. So, let's write the derivative formula using these notations:

As in the analogy with the road, here when the function increases, the derivative is positive, and when it decreases, it is negative.

Is it possible for the derivative to be equal to zero? Certainly. For example, if we are driving on a flat horizontal road, the steepness is zero. And it’s true, the height doesn’t change at all. So it is with the derivative: the derivative of a constant function (constant) is equal to zero:

since the increment of such a function is equal to zero for any.

Let's remember the hilltop example. It turned out that it was possible to arrange the ends of the segment along different sides from the top, so that the height at the ends is the same, that is, the segment is parallel to the axis:

But large segments are a sign of inaccurate measurement. We will raise our segment up parallel to itself, then its length will decrease.

Eventually, when we are infinitely close to the top, the length of the segment will become infinitesimal. But at the same time, it remained parallel to the axis, that is, the difference in heights at its ends is equal to zero (it does not tend to, but is equal to). So the derivative

This can be understood this way: when we stand at the very top, a small shift to the left or right changes our height negligibly.

There is also a purely algebraic explanation: to the left of the vertex the function increases, and to the right it decreases. As we found out earlier, when a function increases, the derivative is positive, and when it decreases, it is negative. But it changes smoothly, without jumps (since the road does not change its slope sharply anywhere). Therefore, between negative and positive values there must definitely be. It will be where the function neither increases nor decreases - at the vertex point.

The same is true for the trough (the area where the function on the left decreases and on the right increases):

A little more about increments.

So we change the argument to magnitude. We change from what value? What has it (the argument) become now? We can choose any point, and now we will dance from it.

Consider a point with a coordinate. The value of the function in it is equal. Then we do the same increment: we increase the coordinate by. What is the argument now? Very easy: . What is the value of the function now? Where the argument goes, so does the function: . What about function increment? Nothing new: this is still the amount by which the function has changed:

Practice finding increments:

1. Find the increment of the function at a point when the increment of the argument is equal to.
2. The same goes for the function at a point.

Solutions:

IN different points with the same argument increment, the function increment will be different. This means that the derivative at each point is different (we discussed this at the very beginning - the steepness of the road is different at different points). Therefore, when we write a derivative, we must indicate at what point:

Power function.

A power function is a function where the argument is to some degree (logical, right?).

Moreover - to any extent: .

The simplest case- this is when the exponent:

Let's find its derivative at a point. Let's recall the definition of a derivative:

So the argument changes from to. What is the increment of the function?

Increment is this. But a function at any point is equal to its argument. That's why:

The derivative is equal to:

The derivative of is equal to:

b) Now consider quadratic function (): .

Now let's remember that. This means that the value of the increment can be neglected, since it is infinitesimal, and therefore insignificant against the background of the other term:

So, we came up with another rule:

c) We continue the logical series: .

This expression can be simplified in different ways: open the first bracket using the formula for abbreviated multiplication of the cube of the sum, or factorize the entire expression using the difference of cubes formula. Try to do it yourself using any of the suggested methods.

So, I got the following:

And again let's remember that. This means that we can neglect all terms containing:

We get: .

d) Similar rules can be obtained for large powers:

e) It turns out that this rule can be generalized for a power function with an arbitrary exponent, not even an integer:

(2)

The rule can be formulated in the words: “the degree is brought forward as a coefficient, and then reduced by .”

We will prove this rule later (almost at the very end). Now let's look at a few examples. Find the derivative of the functions:

1. (in two ways: by formula and using the definition of derivative - by calculating the increment of the function);

1. . Believe it or not, this is a power function. If you have questions like “How is this? Where is the degree?”, remember the topic “”!
   Yes, yes, the root is also a degree, only fractional: .
   So ours Square root- this is just a degree with an indicator:
   .
   We look for the derivative using the recently learned formula:

   If at this point it becomes unclear again, repeat the topic “”!!! (about degree with negative indicator)

2. . Now the exponent:

   And now through the definition (have you forgotten yet?):
   ;
   .
   Now, as usual, we neglect the term containing:
   .

3. . Combination of previous cases: .

Trigonometric functions.

Here we will use one fact from higher mathematics:

With expression.

You will learn the proof in the first year of institute (and to get there, you need to pass the Unified State Exam well). Now I’ll just show it graphically:

We see that when the function does not exist - the point on the graph is cut out. But the closer to the value, the closer the function is to. This is what “aims.”

Additionally, you can check this rule using a calculator. Yes, yes, don’t be shy, take a calculator, we’re not at the Unified State Exam yet.

So, let's try: ;

Don't forget to switch your calculator to Radians mode!

etc. We see that the less, the closer value relationship to

a) Consider the function. As usual, let's find its increment:

Let's turn the difference of sines into a product. To do this, we use the formula (remember the topic “”): .

Now the derivative:

Let's make a replacement: . Then for infinitesimal it is also infinitesimal: . The expression for takes the form:

And now we remember that with the expression. And also, what if an infinitesimal quantity can be neglected in the sum (that is, at).

So we get next rule:the derivative of the sine is equal to the cosine:

These are basic (“tabular”) derivatives. Here they are in one list:

Later we will add a few more to them, but these are the most important, since they are used most often.

Practice:

1. Find the derivative of the function at a point;
2. Find the derivative of the function.

Solutions:

1. First, let's find the derivative in general view, and then substitute its value:
   ;
   .
2. Here we have something similar to power function. Let's try to bring her to
   normal view:
   .
   Great, now you can use the formula:
   .
   .
3. . Eeeeeee.....What is this????

Okay, you're right, we don't yet know how to find such derivatives. Here we have a combination of several types of functions. To work with them, you need to learn a few more rules:

Exponent and natural logarithm.

There is a function in mathematics whose derivative for any value is equal to the value of the function itself at the same time. It is called “exponent”, and is an exponential function

The basis of this function is a constant - it is infinite decimal, that is, an irrational number (such as). It is called the “Euler number”, which is why it is denoted by a letter.

So, the rule:

Very easy to remember.

Well, let’s not go far, let’s look at it right away inverse function. Which function is the inverse of exponential function? Logarithm:

In our case, the base is the number:

Such a logarithm (that is, a logarithm with a base) is called “natural”, and we use a special notation for it: we write instead.

What is it equal to? Of course, .

The derivative of the natural logarithm is also very simple:

Examples:

1. Find the derivative of the function.
2. What is the derivative of the function?

Answers: The exponential and natural logarithm are uniquely simple functions from a derivative perspective. Exponential and logarithmic functions with any other base will have a different derivative, which we will analyze later, after let's go through the rules differentiation.

Rules of differentiation

Rules of what? Again new term, again?!...

Differentiation is the process of finding the derivative.

That's all. What else can you call this process in one word? Not derivative... Mathematicians call the differential the same increment of a function at. This term comes from the Latin differentia - difference. Here.

When deriving all these rules, we will use two functions, for example, and. We will also need formulas for their increments:

There are 5 rules in total.

The constant is taken out of the derivative sign.

If - some constant number (constant), then.

Obviously, this rule also works for the difference: .

Let's prove it. Let it be, or simpler.

Examples.

Find the derivatives of the functions:

1. at a point;
2. at a point;
3. at a point;
4. at the point.

Solutions:

1. (the derivative is the same at all points, since this linear function, remember?);

Derivative of the product

Everything is similar here: let’s enter new feature and find its increment:

Derivative:

Examples:

1. Find the derivatives of the functions and;
2. Find the derivative of the function at a point.

Solutions:

Derivative of an exponential function

Now your knowledge is enough to learn how to find the derivative of any exponential function, and not just exponents (have you forgotten what that is yet?).

So, where is some number.

We already know the derivative of the function, so let's try to reduce our function to a new base:

For this we will use simple rule: . Then:

Well, it worked. Now try to find the derivative, and don't forget that this function is complex.

Happened?

Here, check yourself:

The formula turned out to be very similar to the derivative of an exponent: as it was, it remains the same, only a factor appeared, which is just a number, but not a variable.

Examples:
Find the derivatives of the functions:

Answers:

This is just a number that cannot be calculated without a calculator, that is, it cannot be written down in any more in simple form. Therefore, we leave it in this form in the answer.

Derivative of a logarithmic function

It’s similar here: you already know the derivative of the natural logarithm:

Therefore, to find an arbitrary logarithm with a different base, for example:

We need to reduce this logarithm to the base. How do you change the base of a logarithm? I hope you remember this formula:

Only now we will write instead:

The denominator is simply a constant (a constant number, without a variable). The derivative is obtained very simply:

Derivatives of exponential and logarithmic functions almost never appear in the Unified State Examination, but it wouldn’t hurt to know them.

Derivative of a complex function.

What's happened " complex function"? No, this is not a logarithm, and not an arctangent. These functions can be difficult to understand (although if you find the logarithm difficult, read the topic “Logarithms” and you will be fine), but from a mathematical point of view, the word “complex” does not mean “difficult”.

Imagine a small conveyor belt: two people are sitting and doing some actions with some objects. For example, the first one wraps a chocolate bar in a wrapper, and the second one ties it with a ribbon. The result is a composite object: a chocolate bar wrapped and tied with a ribbon. To eat chocolate, you need to do reverse actions V reverse order.

Let's create a similar mathematical pipeline: first we will find the cosine of a number, and then square the resulting number. So, we are given a number (chocolate), I find its cosine (wrapper), and then you square what I got (tie it with a ribbon). What happened? Function. This is an example of a complex function: when, to find its value, we perform the first action directly with the variable, and then a second action with what resulted from the first.

We can easily do the same steps in reverse order: first you square it, and I then look for the cosine of the resulting number: . It’s easy to guess that the result will almost always be different. Important Feature complex functions: when the order of actions changes, the function changes.

In other words, a complex function is a function whose argument is another function: .

For the first example, .

Second example: (same thing). .

The action we do last will be called "external" function, and the action performed first - accordingly "internal" function(these are informal names, I use them only to explain the material in simple language).

Try to determine for yourself which function is external and which internal:

Answers: Separating inner and outer functions is very similar to changing variables: for example, in a function

1. What action will we perform first? First, let's calculate the sine, and only then cube it. This means that it is an internal function, but an external one.
   And the original function is their composition: .
2. Internal: ; external: .
   Examination: .
3. Internal: ; external: .
   Examination: .
4. Internal: ; external: .
   Examination: .
5. Internal: ; external: .
   Examination: .

We change variables and get a function.

Well, now we will extract our chocolate bar and look for the derivative. The procedure is always reversed: first we look for the derivative of the outer function, then we multiply the result by the derivative of the inner function. In relation to the original example, it looks like this:

Another example:

So, let's finally formulate the official rule:

Algorithm for finding the derivative of a complex function:

It seems simple, right?

Let's check with examples:

Solutions:

1) Internal: ;

External: ;

2) Internal: ;

(Just don’t try to cut it by now! Nothing comes out from under the cosine, remember?)

3) Internal: ;

External: ;

It is immediately clear that this is a three-level complex function: after all, this is already a complex function in itself, and we also extract the root from it, that is, we perform the third action (we put the chocolate in a wrapper and with a ribbon in the briefcase). But there is no reason to be afraid: we will still “unpack” this function in the same order as usual: from the end.

That is, first we differentiate the root, then the cosine, and only then the expression in brackets. And then we multiply it all.

In such cases, it is convenient to number the actions. That is, let's imagine what we know. In what order will we perform actions to calculate the value of this expression? Let's look at an example:

The later the action is performed, the more “external” the corresponding function will be. The sequence of actions is the same as before:

Here the nesting is generally 4-level. Let's determine the course of action.

1. Radical expression. .

2. Root. .

3. Sine. .

4. Square. .

5. Putting it all together:

DERIVATIVE. BRIEFLY ABOUT THE MAIN THINGS

Derivative of a function- the ratio of the increment of the function to the increment of the argument for an infinitesimal increment of the argument:

Basic derivatives:

Rules of differentiation:

The constant is taken out of the derivative sign:

Derivative of the sum:

Derivative of the product:

Derivative of the quotient:

Derivative of a complex function:

Algorithm for finding the derivative of a complex function:

1. We define the “internal” function and find its derivative.
2. We define the “external” function and find its derivative.
3. We multiply the results of the first and second points.

1. argument increment and function increment.

Let the function be given. Let's take two argument values: initial and modified, which is usually denoted
, Where - the amount by which the argument changes when moving from the first value to the second, it is called argument increment.

The argument values ​​and correspond to certain values functions: initial and changed
, magnitude , by which the value of the function changes when the argument changes by value, is called function increment.

2. the concept of the limit of a function at a point.

Number called the limit of the function
with tending to , if for any number
there is such a number
that in front of everyone
, satisfying the inequality
, the inequality will be satisfied
.

Second definition: A number is called the limit of a function as it tends to , if for any number there is a neighborhood of the point such that for any of this neighborhood . Designated
.

3. infinitely large and infinitesimal functions at a point. Endlessly small function at a point - a function whose limit when it tends to a given point equal to zero. Endlessly great function at a point - a function whose limit when it tends to a given point is equal to infinity.

4. main theorems about limits and consequences from them (without proof).

consequence: the constant factor can be taken beyond the limit sign:

If the sequences and converge and the limit of the sequence is nonzero, then

consequence: the constant factor can be taken beyond the limit sign.

11. if there are limits to functions
And
and the limit of the function is non-zero,

then there is also a limit to their relationship, equal to ratio function limits and:

.

12. if
, That
, the converse is also true.

13. Theorem on the limit of an intermediate sequence. If the sequences
converging, and
And
That

5. limit of a function at infinity.

The number a is called the limit of a function at infinity (for x tending to infinity) if for any sequence tending to infinity
corresponds to a sequence of values ​​tending to the number A.

6. limits number sequence.

Number A is called the limit of a number sequence if for any positive number there will be natural number N, such that for all n> N inequality holds
.

Symbolically this is defined as follows:
fair .

The fact that the number A is the limit of the sequence, denoted as follows:

.

7.number "e". natural logarithms.

Number "e" represents the limit of the number sequence, n- th member of which
, i.e.

.

Natural logarithm – logarithm with a base e. natural logarithms are denoted
without specifying a reason.

Number
allows you to move from decimal logarithm to natural and back.

, it is called the transition module from natural logarithms to decimal.

8. wonderful limits
,

.

First wonderful limit:

thus at

by the intermediate sequence limit theorem

second remarkable limit:

.

To prove the existence of a limit
use the lemma: for any real number
And
inequality is true
(2) (at
or
inequality turns into equality.)

Sequence (1) can be written as follows:

.

Now consider an auxiliary sequence with a common term
Let’s make sure that it decreases and is bounded below:
If
, then the sequence decreases. If
, then the sequence is bounded below. Let's show this:

due to equality (2)

i.e.
or
. That is, the sequence is decreasing, and since the sequence is bounded below. If a sequence is decreasing and bounded below, then it has a limit. Then

has a limit and sequence (1), because

And
.

L. Euler called this limit .

9. one-sided limits, discontinuity of function.

number A is the left limit if for any sequence the following holds: .

number A is the right limit if the following holds for any sequence: .

If at the point A belonging to the domain of definition of a function or its boundary, the condition of continuity of the function is violated, then the point A is called a discontinuity point or discontinuity of a function. if, as the point tends

12. sum of terms of infinite decreasing geometric progression. Geometric progression is a sequence in which the ratio between the subsequent and previous terms remains unchanged, this ratio is called the denominator of the progression. Sum of first n members of the geometric progression is expressed by the formula
this formula convenient to use for a decreasing geometric progression - a progression for which absolute value its denominator is less than zero. - first member; - progression denominator; - number of the taken member of the sequence. The sum of an infinite decreasing progression is the number to which the sum of the first terms of a decreasing progression indefinitely approaches when the number increases indefinitely.
That. The sum of the terms of an infinitely decreasing geometric progression is equal to .

Let X– argument (independent variable); y=y(x)– function.

Let's take a fixed argument value x=x 0 and calculate the value of the function y 0 =y(x 0 ) . Now let's arbitrarily set increment (change) of the argument and denote it  X ( X can be of any sign).

Increment argument is a dot X 0 +  X. Let's say it also contains a function value y=y(x 0 +  X)(see picture).

Thus, with an arbitrary change in the value of the argument, a change in the function is obtained, which is called increment function values:

and is not arbitrary, but depends on the type of function and value
.

Argument and function increments can be final, i.e. express oneself constant numbers, in which case they are sometimes called finite differences.

In economics, finite increments are considered quite often. For example, the table shows data on the length of the railway network of a certain state. Obviously, the increment in network length is calculated by subtracting the previous value from the subsequent one.

We will consider the length of the railway network as a function, the argument of which will be time (years).

	Railway length as of December 31, thousand km.	Increment	Average annual growth

In itself, an increase in a function (in this case, the length of the railway network) does not characterize the change in function well. In our example, from the fact that 2,5>0,9 it cannot be concluded that the network grew faster in 2000-2003 years than in 2004 g., because the increment 2,5 refers to a three-year period, and 0,9 - in just one year. Therefore, it is quite natural that an increment in a function leads to a unit change in the argument. The increment of the argument here is periods: 1996-1993=3; 2000-1996=4; 2003-2000=3; 2004-2003=1 .

We get what is called in economic literature average annual growth.

You can avoid the operation of reducing the increment to the unit of argument change if you take the function values ​​for argument values ​​that differ by one, which is not always possible.

In mathematical analysis, in particular in differential calculus, infinitesimal (IM) increments of argument and function are considered.

Differentiation of a function of one variable (derivative and differential) Derivative of a function

Increments of argument and function at a point X 0 can be considered as comparable infinitesimal quantities (see topic 4, comparison of BM), i.e. BM of the same order.

Then their ratio will have a finite limit, which is defined as the derivative of the function in t X 0 .

Limit of the ratio of the increment of a function to the BM increment of the argument at a point x=x 0 called derivative functions at a given point.

The symbolic designation of a derivative by a stroke (or rather, by the Roman numeral I) was introduced by Newton. You can also use a subscript, which shows which variable the derivative is calculated with, for example, . Another notation proposed by the founder of the calculus of derivatives, the German mathematician Leibniz, is also widely used:
. You will learn more about the origin of this designation in the section Function differential and argument differential.

This number estimates speed changes in the function passing through a point
.

Let's install geometric meaning derivative of a function at a point. For this purpose, we will plot the function y=y(x) and mark on it the points that determine the change y(x) in the interim

Tangent to the graph of a function at a point M 0
we will consider the limiting position of the secant M 0 M given that
(dot M slides along the graph of a function to a point M 0 ).

Let's consider
. Obviously,
.

If the point M direct along the graph of the function towards the point M 0 , then the value
will tend to a certain limit, which we denote
. Wherein.

Limit angle  coincides with the angle of inclination of the tangent drawn to the graph of the function incl. M 0 , so the derivative
numerically equal tangent slope at the specified point.

-

geometric meaning of the derivative of a function at a point.

Thus, we can write the tangent and normal equations ( normal - this is a straight line perpendicular to the tangent) to the graph of the function at some point X 0 :

Tangent - .

Normal -
.

Of interest are cases when these lines are located horizontally or vertically (see Topic 3, special cases of the position of a line on a plane). Then,

If
;

If
.

The definition of derivative is called differentiation functions.

 If the function at the point X 0 has a finite derivative, then it is called differentiable at this point. A function that is differentiable at all points of a certain interval is called differentiable on this interval.

 Theorem . If the function y=y(x) differentiable incl. X 0 , then it is continuous at this point.

Thus, continuity– a necessary (but not sufficient) condition for the differentiability of a function.

in medical and biological physics

LECTURE No. 1

DERIVATIVE AND DIFFERENTIAL FUNCTIONS.

PARTIAL DERIVATIVES.

1. The concept of derivative, its mechanical and geometric meaning.

A ) Increment of argument and function.

Let a function y=f(x) be given, where x is the value of the argument from the domain of definition of the function. If you select two values ​​of the argument x o and x from a certain interval of the domain of definition of the function, then the difference between the two values ​​of the argument is called the increment of the argument: x - x o =∆x.

The value of the argument x can be determined through x 0 and its increment: x = x o + ∆x.

The difference between two function values ​​is called the function increment: ∆y =∆f = f(x o +∆x) – f(x o).

The increment of an argument and a function can be represented graphically (Fig. 1). Argument increment and function increment can be either positive or negative. As follows from Fig. 1, geometrically, the increment of the argument ∆х is represented by the increment of the abscissa, and the increment of the function ∆у by the increment of the ordinate. The function increment should be calculated in the following order:

we give the argument an increment ∆x and get the value – x+Δx;

2) find the value of the function for the value of the argument (x+∆x) – f(x+∆x);

3) find the increment of the function ∆f=f(x + ∆x) - f(x).

Example: Determine the increment of the function y=x 2 if the argument changed from x o =1 to x=3. For point x o the value of the function f(x o) = x² o; for the point (x o +∆x) the value of the function f(x o +∆x) = (x o +∆x) 2 = x² o +2x o ∆x+∆x 2, from where ∆f = f(x o + ∆x)–f(x o) = (x o +∆x) 2 –x² o = x² o +2x o ∆x+∆x 2 –x² o = 2x o ∆x+∆x 2; ∆f = 2x o ∆x+∆x 2 ;

∆х = 3–1 = 2; ∆f =2·1·2+4 = 8.b)

Problems leading to the concept of derivative. Definition of derivative, its physical meaning.

The concept of increment of argument and function is necessary to introduce the concept of derivative, which historically arose based on the need to determine the speed of certain processes.

Let's consider how you can determine the speed of rectilinear motion. Let the body move rectilinearly according to the law: ∆S= ·∆t. For uniform motion:= ∆S/∆t. For alternating motion, the value ∆Ѕ/∆t determines the value  avg. , i.e. avg. =∆S/∆t.But average speed

does not make it possible to reflect the features of body movement and give an idea of ​​​​the true speed at time t. When the time period decreases, i.e. at ∆t→0 the average speed tends to its limit – the instantaneous speed:
 instant =
 avg. =

∆S/∆t.

does not make it possible to reflect the features of body movement and give an idea of ​​​​the true speed at time t. When the time period decreases, i.e. at ∆t→0 the average speed tends to its limit – the instantaneous speed:
 instant =
The instantaneous rate of a chemical reaction is determined in the same way:

∆х/∆t, where x is the amount of substance formed during a chemical reaction during time t. Similar tasks

to determine the speed of various processes led to the introduction in mathematics of the concept of a derivative function. Let it be given continuous function
f(x), defined on the interval ]a, in[ie its increment ∆f=f(x+∆x)–f(x).Relation

is a function of ∆x and expresses the average rate of change of the function. Ratio limit :

, when ∆х→0, provided that this limit exists, is called the derivative of the function

.

y" x =
The derivative is denoted: " – (Yigree stroke by X); f ; (x) – (eff prime on x) – y" – (Greek stroke); dy/dх (de igrek by de x);

Based on the definition of the derivative, we can say that the instantaneous speed of rectilinear motion is the time derivative of the path:

 instant = S" t = f " (t).

Thus, we can conclude that the derivative of a function with respect to the argument x is the instantaneous rate of change of the function f(x):

y" x =f " (x)= instant.

This is the physical meaning of the derivative. The process of finding the derivative is called differentiation, so the expression “differentiate a function” is equivalent to the expression “find the derivative of a function.”

V)Geometric meaning of derivative.

P
the derivative of the function y = f(x) has a simple geometric meaning associated with the concept of a tangent to a curved line at some point M. At the same time, tangent, i.e. a straight line is analytically expressed as y = kx = tan· x, where – the angle of inclination of the tangent (straight line) to the X axis. Let us imagine a continuous curve as a function y = f(x), take a point M1 on the curve and a point M1 close to it and draw a secant through them. Her slope k sec =tg β = .If we bring point M 1 closer to M, then the increment in argument ∆х will tend to zero, and the secant at β=α will take the position of a tangent. From Fig. 2 it follows: tgα =
tgβ =
=y" x. But tgα is equal to the slope of the tangent to the graph of the function:

k = tgα =
=y" x = f " (X). So, the angular coefficient of a tangent to the graph of a function at a given point is equal to the value of its derivative at the point of tangency. This is the geometric meaning of the derivative.

G)General rule for finding the derivative.

Based on the definition of the derivative, the process of differentiating a function can be represented as follows:

f(x+∆x) = f(x)+∆f;

find the increment of the function: ∆f= f(x + ∆x) - f(x);

form the ratio of the increment of the function to the increment of the argument:

;

Example: f(x)=x 2 ; " f

(x)=?. However, as can be seen even from this simple example , the application of the specified sequence when taking derivatives is a labor-intensive and complex process. Therefore, for various functions we introduce general formulas

IN differentiation, which are presented in the form of a table of “Basic formulas for differentiation of functions”. coordinate plane xOy consider the graph of the function y=f(x) . Let's fix the point M(x 0 ; f (x 0)) . Let's add an abscissa x 0 incrementΔх . We will get a new abscissa x 0 +Δx N. This is the abscissa of the point , and the ordinate will be equal f (x 0 +Δx ). The change in the abscissa entailed a change in the ordinate. This change is called the function increment and is denoted.

ΔyΔy=f (x 0 +Δx) - f (x 0). Through dots And N M let's draw a secant MN φ , which forms an angle with positive axis direction. Let's determine the tangent of the angle φ from right triangle MPN.

Let increment tends to zero. Then the secant let's draw a secant will tend to take a tangent position MT, and the angle φ will become an angle α . So, the tangent of the angle α is the limiting value of the tangent of the angle φ :

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

Geometric meaning of derivative lies in the fact that the numerical derivative of the function at a given point is equal to the tangent of the angle formed by the tangent drawn through this point to the given curve and the positive direction of the axis with positive axis direction:

Examples.

1. Find the increment of the argument and the increment of the function y= x 2, If initial value the argument was equal 4 , and new - 4,01 .

Solution.

New argument value x=x 0 +Δx. Let's substitute the data: 4.01=4+Δx, hence the increment of the argument increment=4.01-4=0.01. The increment of a function, by definition, is equal to the difference between the new and previous values ​​of the function, i.e. Δy=f (x 0 +Δx) - f (x 0). Since we have a function y=x2, That Δу=(x 0 +Δx) 2 - (x 0) 2 =(x 0) 2 +2x 0 · Δx+(Δx) 2 - (x 0) 2 =2x 0 · Δx+(Δx) 2 =

2 · 4 · 0,01+(0,01) 2 =0,08+0,0001=0,0801.

Answer: argument increment increment=0.01; function increment Δу=0,0801.

The function increment could be found differently: ). The change in the abscissa entailed a change in the ordinate. This change is called the function increment and is denoted=y (x 0 +Δx) -y (x 0)=y(4.01) -y(4)=4.01 2 -4 2 =16.0801-16=0.0801.

2. Find the angle of inclination of the tangent to the graph of the function consider the graph of the function at the point x 0, If f "(x 0) = 1.

Solution.

The value of the derivative at the point of tangency x 0 and is the value of the tangent of the tangent angle (the geometric meaning of the derivative). We have: f "(x 0) = tanα = 1 → α = 45°, because tg45°=1.

Answer: the tangent to the graph of this function forms an angle with the positive direction of the Ox axis equal to 45°.

3. Derive the formula for the derivative of the function y=xn.

Differentiation is the action of finding the derivative of a function.

When finding derivatives, use formulas that were derived based on the definition of a derivative, in the same way as we derived the formula for the derivative degree: (x n)" = nx n-1.

These are the formulas.

Table of derivatives It will be easier to memorize by pronouncing verbal formulations:

1. Derivative constant value equal to zero.

2. X prime is equal to one.

3. Constant multiplier can be taken out of the derivative sign.

4. The derivative of a degree is equal to the product of the exponent of this degree by a degree with the same base, but the exponent is one less.

5. The derivative of a root is equal to one divided by two equal roots.

6. The derivative of one divided by x is equal to minus one divided by x squared.

7. The derivative of the sine is equal to the cosine.

8. The derivative of the cosine is equal to minus sine.

9. The derivative of the tangent is equal to one divided by the square of the cosine.

10. The derivative of the cotangent is equal to minus one divided by the square of the sine.

We teach differentiation rules.

1. The derivative of an algebraic sum is equal to algebraic sum derivatives of terms.

2. The derivative of a product is equal to the product of the derivative of the first factor and the second plus the product of the first factor and the derivative of the second.

3. The derivative of “y” divided by “ve” is equal to a fraction in which the numerator is “y prime multiplied by “ve” minus “y multiplied by ve prime”, and the denominator is “ve squared”.

4. Special case formulas 3.

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