It's very easy to remember.
Well, let's not go far, let's immediately consider inverse function. Which function is the inverse of 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: we 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: Exhibitor and natural logarithm- functions are uniquely simple in terms 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
What rules? Another new term, again?!...
Differentiation is the process of finding the derivative.
Only and everything. What is another word for this process? Not proizvodnovanie... The differential of mathematics is called the very increment of the 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 sign of the derivative.
If - some constant number (constant), then.
Obviously, this rule also works for the difference: .
Let's prove it. Let, or easier.
Examples.
Find derivatives of 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 linear function, remember?);
Derivative of a product
Everything is similar here: we introduce a new function and find its increment:
Derivative:
Examples:
- Find derivatives of functions and;
- Find the derivative of a function at a point.
Solutions:
Derivative of exponential function
Now your knowledge is enough to learn how to find the derivative of any exponential function, and not just the exponent (have you forgotten what it is yet?).
So where is some number.
We already know the derivative of the function, so let's try to bring our function to a new base:
For this we 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 the exponent: as it was, it remains, only a factor appeared, which is just a number, but not a variable.
Examples:
Find derivatives of 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 it is left in this form.
Note that here is the quotient of two functions, so we apply the appropriate differentiation rule:
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 from the logarithm with a different base, for example, :
We need to bring this logarithm to the base. How do you change the base of a logarithm? I hope you remember this formula:
Only now instead of we will write:
The denominator turned out to be just a constant (a constant number, without a variable). The derivative is very simple:
Derivatives of exponential and logarithmic functions almost never occur 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 arc tangent. These functions can be difficult to understand (although if the logarithm seems difficult to you, read the topic "Logarithms" and everything will work out), but in terms of mathematics, the word "complex" does not mean "difficult".
Imagine a small conveyor: two people are sitting and doing some actions with some objects. For example, the first 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 opposite 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, they give us 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, in order to find its value, we perform 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 steps 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 the order of actions changes, the function changes.
Second example: (same). .
The last action we do will be called "external" function, and the action performed 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: The separation of inner and outer functions is very similar to changing variables: for example, in the function
- What action will we take first? First we calculate the sine, and only then we raise it to a cube. So it's an internal function, not 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 - 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. For 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 to be simple, right?
Let's check with examples:
Solutions:
1) Internal: ;
External: ;
2) Internal: ;
(just don’t try to reduce by now! Nothing is taken out from under the cosine, remember?)
3) Internal: ;
External: ;
It is immediately clear that there is a three-level complex function here: after all, this is already a complex function in itself, and we still extract the root from it, that is, we perform the third action (put chocolate in a wrapper and with a ribbon in a briefcase). 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 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 - as before:
Here the nesting is generally 4-level. Let's determine the course of action.
1. Radical expression. .
2. Root. .
3. Sinus. .
4. Square. .
5. Putting it all together:
DERIVATIVE. BRIEFLY ABOUT THE MAIN
Function derivative- the ratio of the increment of the function to the increment of the argument with an infinitesimal increment of the argument:
Basic derivatives:
Differentiation rules:
The constant is taken out of the sign of the derivative:
Derivative of sum:
Derivative product:
Derivative of the quotient:
Derivative of a complex function:
Algorithm for finding the derivative of a complex function:
- We define the "internal" function, find its derivative.
- We define the "external" function, find its derivative.
- We multiply the results of the first and second points.
On which we analyzed the simplest derivatives, and also got acquainted with the rules of differentiation and some techniques finding derivatives. Thus, if you are not very good with 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 easy, 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 in the table at the rule (No. 5) for differentiating a complex function:
We understand. First of all, let's take a look at the notation. Here we have two functions - and , and the function, figuratively speaking, is nested 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 – inner (or nested) function.
! These definitions are not theoretical and should not appear in the final design of assignments. I use the 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 the whole expression, so finding the derivative immediately 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 it is impossible to “tear apart” the sine:
In this example, already from my explanations, it is intuitively clear that the function is a complex function, and the polynomial is an internal function (embedding), and an external function.
First step, which must be performed 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 nested under the sine. But what if it's not obvious? How to determine exactly which function is external and which is internal? To do this, I propose to use the following technique, which can be carried out mentally or on a draft.
Let's imagine that we need to calculate the value of the expression with a calculator (instead of one, there can be any number).
What do we calculate first? Primarily you will need to perform the following action: , so the polynomial will be an internal function: 
Secondly you will need to find, so the sine - will be an external function: 
After we UNDERSTAND with inner and outer functions, it's time to apply the compound function differentiation rule 
.
We start to decide. From the lesson How to find the derivative? we remember that the design of the solution of any derivative always begins like this - we enclose the expression in brackets and put a stroke at the top right:
At first find the derivative of the external function (sine), look at the table of derivatives elementary functions and notice that . All tabular formulas are applicable even if "x" is replaced by 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 
clean looks like this:
The constant factor is usually placed at the beginning of the expression:
If there is any misunderstanding, write down the decision 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: 
We figure out where we have an external function, and where is an internal one. To do this, we try (mentally or on a draft) to calculate the value of the expression for . What needs to 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 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 desired 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: 
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 self-solving (answer at the end of the lesson).
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, reason, where is the external and where is the internal function, why are the tasks solved that way?
Example 5
a) Find the derivative of a 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, we first bring the function into the proper form 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 put the expression in parentheses to common denominator and write it all down as a fraction. It’s beautiful, of course, but when cumbersome long derivatives are obtained, 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 self-solving (answer at the end of the lesson).
It is interesting to note that sometimes, instead of the rule for differentiating a complex function, one can use the rule for differentiating a quotient 
, but such a solution will look like a perversion unusual. 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 take out the minus sign of the derivative, and raise the cosine to the numerator:
Cosine is an internal function, exponentiation is an external function.
Let's use our rule 
:
We find the derivative of the inner 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 self-solving (answer at the end of the lesson).
So far, we have considered 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
We understand the attachments of this function. We try to evaluate the expression using the experimental value . How would we count on a calculator?
First you need to find, which means that the arcsine is the deepest nesting:
This arcsine of unity should then be squared:
And finally, we raise the seven to the power: 
That is, in this example we have three different functions and two nestings, while the innermost function is the arcsine, and the outermost function is the exponential function.
We start to decide
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 of differentiation of a complex function 
next.
It is absolutely impossible to solve physical problems or examples in mathematics without knowledge about the derivative and methods for calculating it. The 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) . The points x and x0 belong to this interval. When x changes, the function itself changes. Argument change - difference of its values x-x0 . This difference is written as delta x and is called argument increment. The change or increment of a function is the difference between the values of the 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 is the point in finding such a limit? But which one:
the derivative of a function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at a given point.
The physical meaning of the derivative: the time derivative of the path is equal to the speed of the rectilinear motion.
Indeed, since school days, everyone knows that speed is a private path. x=f(t) and time t . average speed for some period of time:
To find out the speed of movement at a time t0 you need to calculate the limit:
Rule one: take out the constant
The constant can be taken out of the sign of the derivative. Moreover, it must be done. When solving examples in mathematics, 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: the 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:
Decision: 
Here it is important to say 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 encounter 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 consider the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the intermediate argument itself with respect to the independent variable.
Rule Four: The derivative of the quotient of two functions
Formula for determining the derivative of a quotient of two functions:
We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it seems, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.
With any question on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult control and deal with tasks, even if you have never dealt with the calculation of derivatives before.
Examples of calculating derivatives using the formula for the derivative of a complex function are given.
ContentSee also: Proof of the formula for the derivative of a complex function
Basic formulas
Here we give examples of calculating derivatives of the following functions:
;
;
;
;
.
If a function can be represented as a complex function in the following form:
,
then its derivative is determined by the formula:
.
In the examples below, we will write this formula in the following form:
.
where .
Here, the subscripts or , located under the sign of the derivative, denote the variable with respect to which differentiation is performed.
Usually, in tables of derivatives, the derivatives of functions from the variable x are given. However, x is a formal parameter. The variable x can be replaced by any other variable. Therefore, when differentiating a function from a variable , we simply change, in the table of derivatives, the variable x to the variable u .
Simple examples
Example 1
Find the derivative of a complex function
.
Let's write down given function in equivalent form:
.
In the table of derivatives we find:
;
.
According to the formula for the derivative of a complex function, we have:
.
Here .
Example 2
Find derivative
.
We take out the constant 5 beyond the sign of the derivative and from the table of derivatives we find:
.
.
Here .
Example 3
Find the derivative
.
We take out the constant -1
for the sign of the derivative and from the table of derivatives we find:
;
From the table of derivatives we find:
.
We apply the formula for the derivative of a complex function:
.
Here .
More complex examples
In more difficult examples we apply the rule of differentiation of a complex function several times. In doing so, we calculate the derivative from the end. That is, we break the function into its component parts and find the derivatives of the simplest parts using derivative table. We also apply sum differentiation rules, products and fractions . Then we make substitutions and apply the formula for the derivative of a complex function.
Example 4
Find the derivative
.
Let's single out the most simple part formula and find its derivative. .
.
Here we have used the notation
.
We find the derivative of the next part of the original function, applying the results obtained. We apply the rule of differentiation of the sum:
.
Once again, we apply the rule of differentiation of a complex function.
.
Here .
Example 5
Find the derivative of a function
.
We select the simplest part of the formula and find its derivative from the table of derivatives. .
We apply the rule of differentiation of a complex function.
.
Here
.
We differentiate the next part, applying the results obtained.
.
Here
.
Let's differentiate the next part.
.
Here
.
Now we find the derivative of the desired function.
.
Here
.
complex derivatives. Logarithmic derivative.
Derivative of exponential function
We continue to improve our differentiation technique. In this lesson, we will consolidate the material covered, consider more complex derivatives, and also get acquainted with new tricks and tricks for finding the derivative, in particular, with the logarithmic derivative.
For those readers who low level preparation, refer to the article How to find the derivative? Solution examples which will allow you to raise your skills almost from scratch. Next, you need to carefully study the page Derivative of a compound function, understand and resolve all the examples I have given. This lesson is logically the third in a row, and after mastering it, you will confidently differentiate fairly complex functions. It is undesirable to stick to the position “Where else? And that’s enough!”, since all examples and solutions are taken from real control works and often encountered in practice.
Let's start with repetition. On the lesson Derivative of a compound function we have considered a number of examples with detailed comments. In the course of studying differential calculus and other sections of mathematical analysis, you will have to differentiate very often, and it is not always convenient (and not always necessary) to paint examples in great detail. Therefore, we will practice in the oral finding of derivatives. The most suitable "candidates" for this are derivatives of the simplest of complex functions, for example:
According to the rule of differentiation of a complex function 
:
When studying other topics of matan in the future, such a detailed record is most often not required, it is assumed that the student is able to find similar derivatives on autopilot. Let's imagine that at 3 o'clock in the morning the phone rang and pleasant voice asked: "What is the derivative of the tangent of two x?". This should be followed by an almost instantaneous and polite response: 
.
The first example will be immediately intended for an independent solution.
Example 1
Find the following derivatives orally, in one step, for example: . To complete the task, you only need to use table of derivatives of elementary functions(if she hasn't already remembered). If you have any difficulties, I recommend re-reading the lesson Derivative of a compound function.
, , ,
, 
, , 
, , ,
, , 
,
, , 
,
, , ,
, 
,
Answers at the end of the lesson
Complex derivatives
After preliminary artillery preparation, examples with 3-4-5 attachments of functions will be less scary. Perhaps the following two examples will seem complicated to some, but if they are understood (someone suffers), then almost everything else in differential calculus will seem like a child's joke.
Example 2
Find the derivative of a function 
As already noted, when finding the derivative of a complex function, first of all, it is necessary right UNDERSTAND INVESTMENTS. In cases where there are doubts, I remind you of a useful trick: we take the experimental value "x", for example, and try (mentally or on a draft) to substitute this value into the "terrible expression".
1) First we need to calculate the expression, so the sum is the deepest nesting.
2) Then you need to calculate the logarithm:
4) Then cube the cosine:
5) At the fifth step, the difference:
6) And finally, the outermost function is the square root: 
Complex Function Differentiation Formula 
are applied in reverse order, from the outermost function to the innermost. We decide:
Seems to be no error...
(1) We take the derivative of the square root.
(2) We take the derivative of the difference using the rule 
(3) The derivative of the triple is equal to zero. In the second term, we take the derivative of the degree (cube).
(4) We take the derivative of the cosine.
(5) We take the derivative of the logarithm.
(6) Finally, we take the derivative of the deepest nesting .
It may seem too difficult, but this is not the most brutal example. Take, for example, Kuznetsov's collection and you will appreciate all the charm and simplicity of the analyzed derivative. I noticed that they like to give a similar thing at the exam to check whether the student understands how to find the derivative of a complex function, or does not understand.
The following example is for a standalone solution.
Example 3
Find the derivative of a function
Hint: First we apply the rules of linearity and the rule of differentiation of the product
Full solution and answer at the end of the lesson.
It's time to move on to something more compact and prettier.
It is not uncommon for a situation where the product of not two, but three functions is given in an example. How to find the derivative of the product of three factors?
Example 4
Find the derivative of a function 
First, we look, but is it possible to turn the product of three functions into a product of two functions? For example, if we had two polynomials in the product, then we could open the brackets. But in this example, all functions are different: degree, exponent and logarithm.
In such cases, it is necessary successively apply the product differentiation rule 
twice
The trick is that for "y" we denote the product of two functions: , and for "ve" - the logarithm:. Why can this be done? Is it 
- this is not the product of two factors and the rule does not work?! There is nothing complicated:
Now it remains to apply the rule a second time 
to bracket:
You can still pervert and take something out of the brackets, but in this case it is better to leave the answer in this form - it will be easier to check.
The above example can be solved in the second way: 
Both solutions are absolutely equivalent.
Example 5
Find the derivative of a function
This is an example for an independent solution, in the sample it is solved in the first way.
Consider similar examples with fractions.
Example 6
Find the derivative of a function 
Here you can go in several ways:
Or like this:
But the solution can be written more compactly if, first of all, we use the rule of differentiation of the quotient 
, taking for the whole numerator:
In principle, the example is solved, and if it is left in this form, it will not be a mistake. But if you have time, it is always advisable to check on a draft, but is it possible to simplify the answer? We bring the expression of the numerator to a common denominator and get rid of the three-story fraction:
The disadvantage of additional simplifications is that there is a risk of making a mistake not when finding a derivative, but when banal school transformations. On the other hand, teachers often reject the task and ask to “bring it to mind” the derivative.
A simpler example for a do-it-yourself solution:
Example 7
Find the derivative of a function
We continue to master the techniques for finding the derivative, and now we will consider a typical case when a “terrible” logarithm is proposed for differentiation
Example 8
Find the derivative of a function
Here you can go a long way, using the rule of differentiation of a complex function:
But the very first step immediately plunges you into despondency - you have to take an unpleasant derivative of fractional degree, and then also from the fraction.
So before how to take the derivative of the “fancy” logarithm, it is previously simplified using well-known school properties:
! If you have a practice notebook handy, copy these formulas right there. If you don't have a notebook, draw them on a piece of paper, as the rest of the lesson's examples will revolve around these formulas.
The solution itself can be formulated like this:
Let's transform the function:
We find the derivative: 
The preliminary transformation of the function itself greatly simplified the solution. Thus, when a similar logarithm is proposed for differentiation, it is always advisable to “break it down”.
And now a couple of simple examples for an independent solution:
Example 9
Find the derivative of a function 
Example 10
Find the derivative of a function
All transformations and answers at the end of the lesson.
logarithmic derivative
If the derivative of logarithms is such sweet music, then the question arises, is it possible in some cases to organize the logarithm artificially? Can! And even necessary.
Example 11
Find the derivative of a function 
Similar examples we have recently considered. What to do? One can successively apply the rule of differentiation of the quotient, and then the rule of differentiation of the product. The disadvantage of this method is that you get a huge three-story fraction, which you don’t want to deal with at all.
But in theory and practice there is such a wonderful thing as the logarithmic derivative. Logarithms can be organized artificially by "hanging" them on both sides:
Note
: because function can take negative values, then, generally speaking, you need to use modules: 
, which disappear as a result of differentiation. However, the current design is also acceptable, where by default the complex values. But if with all rigor, then in both cases it is necessary to make a reservation that.
Now you need to “break down” the logarithm of the right side as much as possible (formulas in front of your eyes?). I will describe this process in great detail:
Let's start with the differentiation.
We conclude both parts with a stroke:
The derivative of the right side is quite simple, I will not comment on it, because if you are reading this text, you should be able to handle it with confidence.
What about the left side?
On the left side we have complex function. I foresee the question: “Why, is there one letter “y” under the logarithm?”.
The fact is that this "one letter y" - IS A FUNCTION IN ITSELF(if it is not very clear, refer to the article Derivative of a function implicitly specified). Therefore, the logarithm is an external function, and "y" is an internal function. And we use the compound function differentiation rule 
:
On the left side, as if by magic, we have a derivative. Further, according to the rule of proportion, we throw the “y” from the denominator of the left side to the top of the right side:
And now we remember what kind of "game"-function we talked about when differentiating? Let's look at the condition: 
Final answer: 
Example 12
Find the derivative of a function
This is a do-it-yourself example. Sample design of an example of this type at the end of the lesson.
With the help of the logarithmic derivative, it was possible to solve any of examples No. 4-7, another thing is that the functions there are simpler, and, perhaps, the use of the logarithmic derivative is not very justified.
Derivative of exponential function
We have not considered this function yet. An exponential function is a function that has and the degree and base depend on "x". A classic example that will be given to you in any textbook or at any lecture:
How to find the derivative of an exponential function?
It is necessary to use the technique just considered - the logarithmic derivative. We hang logarithms on both sides:
As a rule, the degree is taken out from under the logarithm on the right side:
As a result, on the right side we have a product of two functions, which will be differentiated according to the standard formula 
.
We find the derivative, for this we enclose both parts under strokes:
The next steps are easy:
Finally: 
If some transformation is not entirely clear, please re-read the explanations of Example #11 carefully.
AT practical tasks exponential function will always be more complicated than the considered lecture example.
Example 13
Find the derivative of a function
We use the logarithmic derivative. 
On the right side we have a constant and the product of two factors - "x" and "logarithm of the logarithm of x" (another logarithm is nested under the logarithm). When differentiating a constant, as we remember, it is better to immediately take it out of the sign of the derivative so that it does not get in the way; and, of course, apply the familiar rule 
:
