The derivative of the function y x c is equal to. Derivative of a complex function
On which we analyzed the simplest derivatives, and also got acquainted with the rules of differentiation and some techniques for finding derivatives. Thus, if you are not very much with the derivatives of functions, or some points of this article are not entirely clear, then first read the above lesson. Please, tune in to a serious mood - the material is not an easy one, but I will still try to present it in a simple and accessible way.
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 in the table at the rule (No. 5) for differentiating a complex function:
Understanding. First of all, let's pay attention to the recording. Here we have two functions - and, moreover, the function, figuratively speaking, is embedded in the function. A function of this kind (when one function is nested within another) is called a complex function.
I will call the function external function and the function - an inner (or nested) function.
! These definitions are not theoretical and should not appear in the final design of the assignments. I use informal expressions "external function", "internal" function only to make it easier for you to understand the material.
In order 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 integer expression, so it will not be possible to find the derivative immediately from the table. 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 you cannot “tear apart” a sine:
In this example, already from my explanations, it is intuitively clear that a function is a complex function, and the polynomial is an internal function (nesting), and an external function.
First step, which must be performed when finding the derivative of a complex function, is that figure out which function is internal and which is external.
In the case of simple examples, it seems clear that a polynomial is nested under the sine. But what if everything is not obvious? How to determine exactly which function is external and which is internal? To do this, I suggest using the following technique, which can be done mentally or on a draft.
Imagine that we need to calculate the value of an 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:, so the polynomial will be an internal function: 
Secondly will need to be found, so sine will be an external function: 
After we Figured out with internal and external functions, it's time to apply the rule of differentiation of a complex function 
.
We start to decide. From the lesson How do I find the derivative? we remember that the design of the solution of any derivative always begins like this - we enclose the expression in parentheses and put a stroke on the top right:
At first find the derivative of the external function (sine), look at the table of derivatives of elementary functions and notice that. All tabular formulas are applicable even if "x" is replaced with a complex expression, in this case:
Note that the inner function has not changed, we do not touch it.
Well, it is quite obvious that
The result of applying the formula 
in the final design it looks like this:
A constant factor is usually placed at the beginning of an expression:
If there is any confusion, write the solution down 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, try (mentally or on a draft) to calculate the value of the expression at. What should be done first? First of all, you need to calculate what the base is equal to: which means that the polynomial is the internal function: 
And, only then the exponentiation is 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 are looking 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 of differentiation of a complex function 
next:
I emphasize again that when we take the derivative of the outer function, the inner function does not change for us: 
Now it remains to find a very simple derivative of the inner function and "comb" the result a little:
Example 4
Find the derivative of a function
This is an example for an independent solution (answer at the end of the tutorial).
To consolidate the understanding of the derivative of a complex function, I will give an example without comments, try to figure it out on your own, speculate where is the external and where is the internal function, why were the tasks 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 degree. Thus, first we bring the function into a form appropriate for differentiation:
Analyzing the function, we come to the conclusion that the sum of three terms is an internal function, and exponentiation is an external function. We apply the rule of differentiation of a complex function 
:
The degree is again represented 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 bring the expression to a common denominator in brackets and write everything down in one fraction. Nice, of course, but when cumbersome long derivatives are obtained, it is better not to do this (it is 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 an independent solution (answer at the end of the tutorial).
It is interesting to note that sometimes, instead of the rule for differentiating a complex function, one can use the rule for differentiating the quotient 
, but such a solution will look unusual as a perversion. Here's a typical example:
Example 8
Find the derivative of a function
Here you can use the rule for differentiating 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 outside the sign of the derivative, and raise the cosine to the numerator:
Cosine is an internal function, exponentiation is an external function.
We use our rule 
:
Find the derivative of the internal function, reset the cosine back down:
Ready. In the considered example, it is important not to get confused in the signs. By the way, try to solve it with the rule 
, the answers must match.
Example 9
Find the derivative of a function
This is an example for an independent solution (answer at the end of the tutorial).
So far, we've looked at cases where we only had one attachment 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. Trying to evaluate the expression using the test value. How would we count on a calculator?
First you need to find, which means that the arcsine is the deepest nesting:
Then this arcsine of one should be squared:
And finally, raise the 7 to the power: 
That is, in this example we have three different functions and two attachments, while the innermost function is the arcsine, and the outermost function is the exponential function.
We start to solve
According to the rule 
first you need to take the derivative of the external 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 of differentiation of a complex function 
next.
It's very easy to remember.
Well, let's not go far, we will immediately consider the inverse function. Which function is the inverse of the exponential function? Logarithm:
In our case, the base is a number:
Such a logarithm (that is, a logarithm with a base) is called "natural", and we use a special notation for it: write instead.
What is equal to? Of course, .
The derivative of the natural logarithm is also very simple:
Examples:
- Find the derivative of the function.
- What is the derivative of the function?
Answers: The exponent and natural logarithm are uniquely simple functions from the point of view of the derivative. 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.
Differentiation rules
The rules of what? Again a new term, again ?! ...
Differentiation is the process of finding a derivative.
That's all. How else to call this process in one word? Not a derivation ... The differential of mathematics is called 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 also need formulas for their increments:
There are 5 rules in total.
The constant is moved outside the derivative sign.
If is some constant number (constant), then.
Obviously, this rule also works for the difference:.
Let's prove it. Let, or easier.
Examples.
Find the derivatives of the functions:
- at the point;
- at the point;
- at the point;
- at the point.
Solutions:
- (the derivative is the same at all points, since it is a linear function, remember?);
Derivative of a work
Everything is the same here: we introduce a new function and find its increment:
Derivative:
Examples:
- Find the derivatives of the functions and;
- Find the derivative of the function at the point.
Solutions:
Derivative of the exponential function
Now your knowledge is enough to learn how to find the derivative of any exponential function, not just the exponent (have you forgotten what it is?).
So, where is some number.
We already know the derivative of the function, so let's try to cast our function to a new radix:
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 tricky.
Happened?
Here, check yourself:
The formula turned out to be very similar to the derivative of the exponent: as it was, it remains, only a multiplier 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 in a simpler form. Therefore, in the answer we leave it in this form.
Note that here is the quotient of two functions, so we apply the corresponding rule of differentiation:
In this example, the product of two functions:
Derivative of a logarithmic function
Here it is similar: you already know the derivative of the natural logarithm:
Therefore, to find an arbitrary one of the logarithm with a different base, for example:
You need to bring this logarithm to the base. How do you change the base of the logarithm? I hope you remember this formula:
Only now, instead of we will write:
The denominator is just a constant (constant number, no variable). The derivative is very simple:
The derivatives of exponential and logarithmic functions are almost never found in the exam, 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 the logarithm seems difficult to you, read the topic "Logarithms" and everything will pass), but from the point of view of mathematics, the word "difficult" does not mean "difficult".
Imagine a small conveyor belt: two people are sitting and doing some kind of action with some objects. For example, the first one wraps a chocolate bar in a wrapper, and the second ties it with a ribbon. It turns out such 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 we will square the resulting number. So, we are given a number (chocolate bar), I find its cosine (wrapper), and then you square what I have (you tie it with a ribbon). What happened? Function. This is an example of a complex function: when, to find its value, we do the first action directly with the variable, and then another second action with the result of the first.
In other words, a complex function is a function whose argument is another function: .
For our example,.
We may well do the same actions in reverse order: first you square, and then I look for the cosine of the resulting number:. It is easy to guess that the result will almost always be different. An important feature of complex functions: when you change the order of actions, the function changes.
Second example: (same). ...
The action that we do last will be called "External" function, and the action taken first - respectively "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 is internal:
Answers: Separating inner and outer functions is very similar to changing variables: for example, in a function
- What is the first action to take? First, we will calculate the sine, and only then will we raise it to a cube. This means that it is an internal function, but an external one.
And the original function is their composition:. - Internal:; external:.
Examination: . - Internal:; external:.
Examination: . - Internal:; external:.
Examination: . - Internal:; external:.
Examination: .
we change variables and get a function.
Well, now we will extract our chocolate bar - look for a 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 us finally formulate an official rule:
Algorithm for finding the derivative of a complex function:
Everything seems to be simple, right?
Let's check with examples:
Solutions:
1) Internal:;
External:;
2) Internal:;
(just do not try to reduce by now! Nothing can be taken 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 from it we also extract the root, that is, we perform the third action (we put a chocolate bar in a wrapper and put it in a briefcase with a ribbon). But there is no reason to be afraid: anyway, we will "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 all this.
In such cases, it is convenient to number the steps. That is, let's imagine what we know. In what order will we perform actions to calculate the value of this expression? Let's take an example:
The later the action is performed, the more “external” the corresponding function will be. The sequence of actions - as before:
Here nesting is generally 4-level. Let's define a course of action.
1. A radical expression. ...
2. Root. ...
3. Sinus. ...
4. Square. ...
5. Putting everything together:
DERIVATIVE. BRIEFLY ABOUT THE MAIN
Derivative of a function- the ratio of the increment of the function to the increment of the argument with an infinitely small increment of the argument:
Basic derivatives:
Differentiation rules:
The constant is moved outside the derivative sign:
Derivative of the amount:
Derivative of the work:
Derivative of the quotient:
Derivative of a complex function:
Algorithm for finding the derivative of a complex function:
- We define the "internal" function, we find its derivative.
- We define the "external" function, we find its derivative.
- We multiply the results of the first and second points.
Probably, the concept of a derivative is familiar to each of us since school. Usually, students have difficulty understanding this, undoubtedly, very important thing. It is actively used in various areas of human life, and many engineering developments were based precisely on mathematical calculations obtained using a derivative. But before moving on to an analysis of what the derivatives of numbers are, how to calculate them, and where they come in handy, let's plunge a little into history.
History
The basis of mathematical analysis was discovered (it is better to even say "invented", because it did not exist in nature as such) by Isaac Newton, whom we all know from the discovery of the law of universal gravitation. It was he who first applied this concept in physics to link the nature of the speed and acceleration of bodies. And many scientists still praise Newton for this magnificent invention, because in fact he invented the basis of differential and integral calculus, in fact, the basis of a whole field of mathematics called "mathematical analysis". Had the Nobel Prize been at that time, Newton would most likely have received it several times.
Not without other great minds. In addition to Newton, such eminent geniuses of mathematics as Leonard Euler, Louis Lagrange and Gottfried Leibniz worked on the development of the derivative and the integral. It is thanks to them that we got the theory in the form in which it exists to this day. By the way, it was Leibniz who discovered the geometric meaning of the derivative, which turned out to be nothing more than the tangent of the angle of inclination of the tangent to the graph of the function.
What are derivatives of numbers? Let's repeat a little what we went through at school.
What is a derivative?
This concept can be defined in several different ways. The simplest explanation: a derivative is the rate of change of a function. Imagine a graph of some function y versus x. If it is not a straight line, then it has some bends in the graph, periods of increasing and decreasing. If we take any infinitesimal interval of this graph, it will be a straight line segment. So, the ratio of the size of this infinitesimal segment along the y coordinate to the size along the x coordinate will be the derivative of this function at a given point. If we consider the function as a whole, and not at a specific point, then we get the function of the derivative, that is, a certain dependence of the game on x.
In addition, besides the rate of change of the function, there is also a geometric meaning. We will talk about him now.
Geometric meaning
Derivatives of numbers themselves represent a certain number, which, without proper understanding, does not carry any meaning. It turns out that the derivative not only shows the rate of growth or decrease of the function, but also the tangent of the slope of the tangent to the graph of the function at a given point. Not entirely clear definition. Let's analyze it in more detail. Let's say we have a graph of some function (let's take a curve for interest). There are an infinite number of points on it, but there are areas where only one single point has a maximum or minimum. Through any such point, you can draw a straight line that would be perpendicular to the graph of the function at this point. Such a line will be called a tangent line. Let's say we have drawn it to the intersection with the OX axis. So, the angle obtained between the tangent and the OX axis will be determined by the derivative. More precisely, the tangent of this angle will be equal to it.
Let's talk a little about special cases and analyze the derivatives of numbers.
Special cases
As we said, derivatives of numbers are the values of the derivative at a particular point. For example, let's take the function y = x 2. The derivative x is a number, and in the general case is a function equal to 2 * x. If we need to calculate the derivative, say, at the point x 0 = 1, then we get y "(1) = 2 * 1 = 2. Everything is very simple. An interesting case is the derivative. Let's just say that this number, which contains the so-called imaginary unit - a number whose square is equal to 1. Calculation of such a derivative is possible only if the following conditions are met:
1) There must be first-order partial derivatives of the real and imaginary parts in terms of y and x.
2) The Cauchy-Riemann conditions are satisfied, which are related to the equality of partial derivatives described in the first paragraph.
Another interesting case, although not as difficult as the previous one, is the derivative of a negative number. In fact, any negative number can be thought of as a positive number multiplied by -1. Well, the derivative of the constant and the function is equal to the constant multiplied by the derivative of the function.
It will be interesting to learn about the role of the derivative in everyday life, and this is what we will discuss now.
Application
Probably, each of us at least once in his life catches himself thinking that mathematics is unlikely to be useful to him. And such a complex thing as a derivative probably has no application at all. In fact, mathematics - and all its fruits are developed mainly by physics, chemistry, astronomy and even economics. The derivative laid the foundation which gave us the ability to draw conclusions from the graphs of functions, and we learned to interpret the laws of nature and turn them in our favor thanks to him.
Conclusion
Of course, not everyone may need a derivative in real life. But mathematics develops the logic that will certainly be needed. It is not for nothing that mathematics is called the queen of sciences: the foundations for understanding other areas of knowledge are formed from it.
It is absolutely impossible to solve physical problems or examples in mathematics without knowledge of the derivative and the methods of calculating it. Derivative is one of the most important concepts of 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 the derivative
Let there be a function f (x) given in some interval (a, b) ... Points х and х0 belong to this interval. When x changes, the function itself changes. Changing an argument - the difference between 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 in the values of a function at two points. Derivative definition:
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 in finding such a limit? And here's what:
the derivative of the 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 this point.
The physical meaning of the derivative: the derivative of the path with respect to time is equal to the speed of the rectilinear motion.
Indeed, since school times, everyone knows that speed is a private path. x = f (t) and time t ... Average speed over a period of time:
To find out the speed of movement at a time t0 you need to calculate the limit:
Rule one: take out a constant
The constant can be moved outside the sign of the derivative. Moreover, it must be done. When solving examples in math, take as a rule - if you can simplify the expression, be sure to simplify .
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 a 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 say here about the calculation of derivatives of complex functions. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument by the derivative of the intermediate argument with respect to the independent variable.
In the above example, we meet 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 immediate intermediate argument with respect to the independent variable.
Rule four: the quotient derivative of two functions
Formula for determining the derivative of the quotient of two functions:
We tried to tell you about derivatives for dummies from scratch. This topic is not as simple as it sounds, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.
With any question on this and other topics, you can contact student service... In a short time, we will help you solve the most difficult test and deal with tasks, even if you have never done calculating derivatives before.
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