10 derivative of a complex function. Complex derivatives

How to find the derivative, how to take the derivative? In this lesson we will learn how to find derivatives of functions. But before studying this page, I strongly recommend that you familiarize yourself with the methodological material Hot formulas for school mathematics course. The reference manual can be opened or downloaded on the page Mathematical formulas and tables. Also from there we will need Derivatives table, it is better to print it out; you will often have to refer to it, not only now, but also offline.

Eat? Let's get started. I have two news for you: good and very good. The good news is this: to learn how to find derivatives, you don’t have to know and understand what a derivative is. Moreover, it is more expedient to digest the definition of the derivative of a function, the mathematical, physical, geometric meaning of the derivative later, since a high-quality study of the theory, in my opinion, requires the study of a number of other topics, as well as some practical experience.
And now our task is to master these same derivatives technically. The very good news is that learning to take derivatives is not so difficult; there is a fairly clear algorithm for solving (and explaining) this task; integrals or limits, for example, are more difficult to master.

I recommend the following order of studying the topic:: First, this article. Then you need to read the most important lesson Derivative of a complex function. These two basic classes will take your skills from scratch. Next you can get acquainted with more complex derivatives in the article Complex derivatives. Logarithmic derivative. If the bar is too high, read the thing first The simplest typical problems with derivatives. In addition to the new material, the lesson covers other, simpler types of derivatives, and is a great opportunity to improve your differentiation technique. In addition, test papers almost always contain tasks on finding derivatives of functions that are specified implicitly or parametrically. There is also such a lesson: Derivatives of implicit and parametrically defined functions.

I will try in an accessible form, step by step, to teach you how to find derivatives of functions. All information is presented in detail, in simple words.

Actually, let’s immediately look at an example:

Example 1

Find the derivative of a function

Solution:

This is a simple example, please find it in the table of derivatives of elementary functions. Now let's look at the solution and analyze what happened? And the following thing happened: we had a function, which, as a result of the solution, turned into a function.

To put it quite simply, in order to find the derivative of a function, you need to turn it into another function according to certain rules. Look again at the table of derivatives - there functions turn into other functions. The only exception is the exponential function, which turns into itself. The operation of finding the derivative is called differentiation .

Designations: The derivative is denoted by or .

ATTENTION, IMPORTANT! Forgetting to put a stroke (where it is necessary), or to draw an extra stroke (where it is not necessary) - BIG MISTAKE! A function and its derivative are two different functions!

Let's return to our table of derivatives. From this table it is desirable memorize: rules of differentiation and derivatives of some elementary functions, especially:

derivative of the constant:
, where is a constant number;

derivative of a power function:
, in particular: , , .

Why remember? This knowledge is basic knowledge about derivatives. And if you cannot answer the teacher’s question “What is the derivative of a number?”, then your studies at the university may end for you (I am personally familiar with two real life cases). In addition, these are the most common formulas that we have to use almost every time we come across derivatives.

In reality, simple tabular examples are rare; usually, when finding derivatives, differentiation rules are first used, and then a table of derivatives of elementary functions.

In this regard, we move on to consider differentiation rules:

1) A constant number can (and should) be taken out of the derivative sign

Where is a constant number (constant)

Example 2

Find the derivative of a function

Let's look at the table of derivatives. The derivative of the cosine is there, but we have .

It's time to use the rule, we take the constant factor out of the sign of the derivative:

Now we convert our cosine according to the table:

Well, it’s advisable to “comb” the result a little - put the minus sign in first place, at the same time getting rid of the brackets:

2) The derivative of the sum is equal to the sum of the derivatives

Example 3

Find the derivative of a function

Let's decide. As you probably already noticed, the first step that is always performed when finding a derivative is that we enclose the entire expression in parentheses and put a prime at the top right:

Let's apply the second rule:

Please note that for differentiation, all roots and degrees must be represented in the form, and if they are in the denominator, then move them up. How to do this is discussed in my teaching materials.

Now let’s remember the first rule of differentiation - we take the constant factors (numbers) outside the sign of the derivative:

Usually, during the solution, these two rules are applied simultaneously (so as not to rewrite a long expression again).

All functions located under the strokes are elementary table functions; using the table we carry out the transformation:

You can leave everything as is, since there are no more strokes, and the derivative has been found. However, expressions like this usually simplify:

It is advisable to represent all powers of the type again in the form of roots; powers with negative exponents should be reset to the denominator. Although you don't have to do this, it won't be a mistake.

Example 4

Find the derivative of a function

Try to solve this example yourself (answer at the end of the lesson). Those interested can also use intensive course in pdf format, which is especially relevant if you have very little time at your disposal.

3) Derivative of the product of functions

It seems that the analogy suggests the formula ...., but the surprise is that:

This is an unusual rule (as, in fact, others) follows from derivative definitions. But we’ll hold off on the theory for now - now it’s more important to learn how to solve:

Example 5

Find the derivative of a function

Here we have the product of two functions depending on .
First we apply our strange rule, and then we transform the functions using the derivative table:

Difficult? Not at all, quite accessible even for a teapot.

Example 6

Find the derivative of a function

This function contains the sum and product of two functions - quadratic trinomial and logarithm. From school we remember that multiplication and division take precedence over addition and subtraction.

It's the same here. AT FIRST we use the product differentiation rule:

Now for the bracket we use the first two rules:

As a result of applying the rules of differentiation under the strokes, we are left with only elementary functions; using the table of derivatives, we transform them into other functions:

Ready.

With some experience in finding derivatives, simple derivatives do not seem to need to be described in such detail. In general, they are usually decided orally, and it is immediately written down that .

Example 7

Find the derivative of a function

This is an example for you to solve on your own (answer at the end of the lesson)

4) Derivative of quotient functions

A hatch opened in the ceiling, don't be alarmed, it's a glitch.
But this is the harsh reality:

Example 8

Find the derivative of a function

What’s missing here – sum, difference, product, fraction…. What should I start with?! There are doubts, there are no doubts, but, ANYWAY First, draw brackets and put a stroke at the top right:

Now we look at the expression in brackets, how can we simplify it? In this case, we notice a factor, which, according to the first rule, it is advisable to place outside the sign of the derivative.

• Table of derivatives of exponential and logarithmic functions

Derivatives of simple functions

Explanation:
The derivative shows the rate at which the value of a function changes when its argument changes. Since the number does not change in any way under any conditions, the rate of its change is always zero.

2. Derivative of a variable equal to one
x´ = 1

Explanation:
With each increment of the argument (x) by one, the value of the function (the result of the calculation) increases by the same amount. Thus, the rate of change in the value of the function y = x is exactly equal to the rate of change in the value of the argument.

3. The derivative of a variable and a factor is equal to this factor
сx´ = с
Example:
(3x)´ = 3
(2x)´ = 2
Explanation:
In this case, every time the function argument changes ( X) its value (y) increases in With once. Thus, the rate of change of the function value in relation to the rate of change of the argument is exactly equal to the value With.

Whence it follows that
(cx + b)" = c
that is, the differential of the linear function y=kx+b is equal to the slope of the line (k).

5. Derivative of a variable to a power equal to the product of a number of this power and a variable to the power reduced by one
(x c)"= cx c-1, provided that x c and cx c-1 are defined and c ≠ 0
Example:
(x 2)" = 2x
(x 3)" = 3x 2
To remember the formula:
Move the degree of the variable down as a factor, and then reduce the degree itself by one. For example, for x 2 - the two was ahead of the x, and then the reduced power (2-1 = 1) simply gave us 2x. The same thing happened for x 3 - we “move down” the triple, reduce it by one and instead of a cube we have a square, that is, 3x 2. A little "unscientific" but very easy to remember.

6.Derivative of a fraction 1/x
(1/x)" = - 1 / x 2
Example:
Since a fraction can be represented as raising to a negative power
(1/x)" = (x -1)", then you can apply the formula from rule 5 of the table of derivatives
(x -1)" = -1x -2 = - 1 / x 2

7. Derivative of a fraction with a variable of arbitrary degree in the denominator
(1 / x c)" = - c / x c+1
Example:
(1 / x 2)" = - 2 / x 3

8. Derivative of the root(derivative of variable under square root)
(√x)" = 1 / (2√x) or 1/2 x -1/2
Example:
(√x)" = (x 1/2)" means you can apply the formula from rule 5
(x 1/2)" = 1/2 x -1/2 = 1 / (2√x)

9. Derivative of a variable under the root of an arbitrary degree
(n √x)" = 1 / (n n √x n-1)

First level

Derivative of a function. The Ultimate 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. Indeed, on different sections of the road, moving forward (along the x-axis) by one kilometer, we will rise or fall by a different number of meters relative to sea level (along the y-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 the 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 the end point is lower than the starting point, 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:

Let us assume that on some section of the road, when moving forward by a kilometer, the road rises up by a kilometer. Then the slope at this place is equal. And if the road, while moving forward by m, dropped by km? Then the slope is equal.

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. It is necessary to consider smaller areas for a more adequate and accurate assessment of 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 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 is even greater than what happens. 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 a completely ordinary 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.

Can the derivative 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 on opposite sides of the vertex in such a way that the height at the ends turns out to be 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 height difference 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, there must be between negative and positive values. 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:

At 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 is when the exponent is:

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 the 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:

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: .
   This means that our square root is just a power with an exponent:
   .
   We look for the derivative using the recently learned formula:

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

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 smaller, the closer the value of the ratio 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 the following 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 form, and then substitute its value:
   ;
   .
2. Here we have something similar to a 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 base of this function - a constant - is an infinite decimal fraction, 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 immediately consider the inverse function. Which function is the inverse of the 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 we go through the rules of differentiation.

Rules of differentiation

Rules of what? Again a 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 it is a linear function, remember?);

Derivative of the product

Everything is similar here: let’s introduce a new function 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:

To do this, we will use a 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 a simpler 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 are almost never found in the Unified State Examination, but it will not be superfluous to know them.

Derivative of a complex function.

What is a "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 a chocolate bar, you need to do the reverse steps in 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. An important feature of 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.

On which we examined the simplest derivatives, and also became acquainted with the rules of differentiation and some technical techniques for finding derivatives. Thus, if you are not very good with derivatives of functions or some points in this article are not entirely clear, then first read the above lesson. Please get in a serious mood - the material is not simple, but I will still try to present it simply and clearly.

In practice, you have to deal with the derivative of a complex function very often, I would even say, almost always, when you are given tasks to find derivatives.

We look at the table at the rule (No. 5) for differentiating a complex function:

Let's figure it out. First of all, let's pay attention to the entry. Here we have two functions - and , and the function, figuratively speaking, is nested within the function . A function of this type (when one function is nested within another) is called a complex function.

I will call the function external function, and the function – internal (or nested) function.

! These definitions are not theoretical and should not appear in the final design of assignments. I use informal expressions “external function”, “internal” function only to make it easier for you to understand the material.

To clarify the situation, consider:

Example 1

Find the derivative of a function

Under the sine we have not just the letter “X”, but an entire expression, so finding the derivative right away from the table will not work. We also notice that it is impossible to apply the first four rules here, there seems to be a difference, but the fact is that the sine cannot be “torn into pieces”:

In this example, it is already intuitively clear from my explanations that a function is a complex function, and the polynomial is an internal function (embedding), and an external function.

First step what you need to do when finding the derivative of a complex function is to understand which function is internal and which is external.

In the case of simple examples, it seems clear that a polynomial is embedded under the sine. But what if everything is not obvious? How to accurately determine which function is external and which is internal? To do this, I suggest using the following technique, which can be done mentally or in a draft.

Let's imagine that we need to calculate the value of the expression at on a calculator (instead of one there can be any number).

What will we calculate first? First of all you will need to perform the following action: , therefore the polynomial will be an internal function:

Secondly will need to be found, so sine – will be an external function:

After we SOLD OUT with internal and external functions, it’s time to apply the rule of differentiation of complex functions .

Let's start deciding. From the lesson How to find the derivative? we remember that the design of a solution to any derivative always begins like this - we enclose the expression in brackets and put a stroke at the top right:

At first we find the derivative of the external function (sine), look at the table of derivatives of elementary functions and notice that . All table formulas are also applicable if “x” is replaced with a complex expression, in this case:

Please note that the inner function hasn't changed, we don't touch it.

Well, it's quite obvious that

The result of applying the formula in its final form it looks like this:

The constant factor is usually placed at the beginning of the expression:

If there is any misunderstanding, write the solution down on paper and read the explanations again.

Example 2

Find the derivative of a function

Example 3

Find the derivative of a function

As always, we write down:

Let's figure out where we have an external function and where we have an internal one. To do this, we try (mentally or in a draft) to calculate the value of the expression at . What should you do first? First of all, you need to calculate what the base is equal to: therefore, the polynomial is the internal function:

And, only then is the exponentiation performed, therefore, the power function is an external function:

According to the formula , first you need to find the derivative of the external function, in this case, the degree. We look for the required formula in the table: . We repeat again: any tabular formula is valid not only for “X”, but also for a complex expression. Thus, the result of applying the rule for differentiating a complex function next:

I emphasize again that when we take the derivative of the external function, our internal function does not change:

Now all that remains is to find a very simple derivative of the internal function and tweak the result a little:

Example 4

Find the derivative of a function

This is an example for you to solve on your own (answer at the end of the lesson).

To consolidate your understanding of the derivative of a complex function, I will give an example without comments, try to figure it out on your own, reason where the external and where the internal function is, why the tasks are solved this way?

Example 5

a) Find the derivative of the function

b) Find the derivative of the function

Example 6

Find the derivative of a function

Here we have a root, and in order to differentiate the root, it must be represented as a power. Thus, first we bring the function into the form appropriate for differentiation:

Analyzing the function, we come to the conclusion that the sum of the three terms is an internal function, and raising to a power is an external function. We apply the rule of differentiation of complex functions :

We again represent the degree as a radical (root), and for the derivative of the internal function we apply a simple rule for differentiating the sum:

Ready. You can also reduce the expression to a common denominator in brackets and write everything down as one fraction. It’s beautiful, of course, but when you get cumbersome long derivatives, it’s better not to do this (it’s easy to get confused, make an unnecessary mistake, and it will be inconvenient for the teacher to check).

Example 7

Find the derivative of a function

This is an example for you to solve on your own (answer at the end of the lesson).

It is interesting to note that sometimes instead of the rule for differentiating a complex function, you can use the rule for differentiating a quotient , but such a solution will look like an unusual perversion. Here is a typical example:

Example 8

Find the derivative of a function

Here you can use the rule of differentiation of the quotient , but it is much more profitable to find the derivative through the rule of differentiation of a complex function:

We prepare the function for differentiation - we move the minus out of the derivative sign, and raise the cosine into the numerator:

Cosine is an internal function, exponentiation is an external function.
Let's use our rule :

We find the derivative of the internal function and reset the cosine back down:

Ready. In the example considered, it is important not to get confused in the signs. By the way, try to solve it using the rule , the answers must match.

Example 9

Find the derivative of a function

This is an example for you to solve on your own (answer at the end of the lesson).

So far we have looked at cases where we had only one nesting in a complex function. In practical tasks, you can often find derivatives, where, like nesting dolls, one inside the other, 3 or even 4-5 functions are nested at once.

Example 10

Find the derivative of a function

Let's understand the attachments of this function. Let's try to calculate the expression using the experimental value. How would we count on a calculator?

First you need to find , which means the arcsine is the deepest embedding:

This arcsine of one should then be squared:

And finally, we raise seven to a power:

That is, in this example we have three different functions and two embeddings, while the innermost function is the arcsine, and the outermost function is the exponential function.

Let's start deciding

According to the rule First you need to take the derivative of the outer function. We look at the table of derivatives and find the derivative of the exponential function: The only difference is that instead of “x” we have a complex expression, which does not negate the validity of this formula. So, the result of applying the rule for differentiating a complex function next.