Proof and derivation of formulas for the derivative of the natural logarithm and the base a logarithm. Examples of calculating derivatives of ln 2x, ln 3x and ln nx. Proof of the formula for the derivative of the n-th order of the logarithm by the method of mathematical induction.

Content

See also: Logarithm - properties, formulas, graph
Natural logarithm - properties, formulas, graph

Derivation of formulas for derivatives of the natural logarithm and the logarithm base a

The derivative of the natural logarithm of x is equal to one divided by x:
(1) (ln x) ′ =.

The derivative of the logarithm base a is equal to one divided by the variable x times the natural logarithm of a:
(2) (log a x) ′ =.

Proof

Let there be some positive number that is not equal to one. Consider a function that depends on the variable x, which is the logarithm to the base:
.
This function is defined at. Let us find its derivative with respect to the variable x. By definition, the derivative is the following limit:
(3) .

Let's transform this expression to reduce it to the well-known mathematical properties and rules. To do this, we need to know the following facts:
A) Logarithm properties. We need the following formulas:
(4) ;
(5) ;
(6) ;
B) Continuity of the logarithm and the property of limits for a continuous function:
(7) .
Here is some function that has a limit and this limit is positive.
V) The meaning of the second remarkable limit:
(8) .

We apply these facts to our limit. First, we transform the algebraic expression
.
For this we apply properties (4) and (5).

.

Let us use property (7) and the second remarkable limit (8):
.

And finally, we apply property (6):
.
Logarithm base e called natural logarithm... It is designated as follows:
.
Then ;
.

Thus, we have obtained formula (2) for the derivative of the logarithm.

Derivative of the natural logarithm

Once again, we write out the formula for the derivative of the logarithm with respect to the base a:
.
This formula has the simplest form for the natural logarithm, for which,. Then
(1) .

Because of this simplicity, the natural logarithm is very widely used in mathematical analysis and in other branches of mathematics related to differential calculus. Logarithmic functions with other bases can be expressed in terms of the natural logarithm using property (6):
.

The base derivative of the logarithm can be found from formula (1), if the constant is taken out of the differentiation sign:
.

Other ways to prove the derivative of the logarithm

Here we assume that we know the formula for the derivative of the exponent:
(9) .
Then we can derive the formula for the derivative of the natural logarithm, given that the logarithm is the inverse of the exponential function.

Let us prove the formula for the derivative of the natural logarithm, by applying the formula for the derivative of the inverse function:
.
In our case . The function inverse to the natural logarithm is the exponent:
.
Its derivative is determined by formula (9). Variables can be designated with any letter. In formula (9), replace the variable x with y:
.
Since, then
.
Then
.
The formula is proven.

Now we prove the formula for the derivative of the natural logarithm using complex function differentiation rules... Since the functions and are inverse to each other, then
.
We differentiate this equation with respect to the variable x:
(10) .
The x-derivative is equal to one:
.
We apply the rule of differentiating a complex function:
.
Here . Substitute in (10):
.
From here
.

Example

Find derivatives of ln 2x, ln 3x and ln nx.

The original functions are similar. Therefore, we will find the derivative of the function y = ln nx... Then plug in n = 2 and n = 3. And, thus, we obtain formulas for the derivatives of ln 2x and ln 3x .

So, we are looking for the derivative of the function
y = ln nx .
Let's imagine this function as a complex function, consisting of two functions:
1) Variable-dependent functions:;
2) Variable-dependent functions:.
Then the original function is composed of functions and:
.

Let us find the derivative of the function with respect to the variable x:
.
Let's find the derivative of the function with respect to the variable:
.
We apply the formula for the derivative of a complex function.
.
Here we set up.

So we found:
(11) .
We see that the derivative is independent of n. This result is quite natural if we transform the original function using the formula for the logarithm of the product:
.
is constant. Its derivative is zero. Then, according to the rule for differentiating the sum, we have:
.

; ; .

Derivative of the logarithm of the modulus x

Let's find the derivative of another very important function - the natural logarithm of the modulus x:
(12) .

Let's consider a case. Then the function has the form:
.
Its derivative is determined by the formula (1):
.

Now consider the case. Then the function has the form:
,
where .
But we also found the derivative of this function in the above example. It does not depend on n and is equal to
.
Then
.

We combine these two cases into one formula:
.

Accordingly, for the logarithm base a, we have:
.

Higher order derivatives of the natural logarithm

Consider the function
.
We found its first-order derivative:
(13) .

Find the second-order derivative:
.
Find the third-order derivative:
.
Let's find the derivative of the fourth order:
.

It can be seen that the nth order derivative has the form:
(14) .
Let us prove this by the method of mathematical induction.

Proof

Let us substitute the value n = 1 into formula (14):
.
Since, then for n = 1 , formula (14) is valid.

Suppose that formula (14) holds for n = k. Let us prove that this implies that the formula is valid for n = k + 1 .

Indeed, for n = k we have:
.
We differentiate with respect to the variable x:

.
So we got:
.
This formula coincides with formula (14) for n = k + 1 ... Thus, from the assumption that formula (14) is valid for n = k, it follows that formula (14) is valid for n = k + 1 .

Therefore, formula (14), for the derivative of the nth order, is valid for any n.

Higher order derivatives of the logarithm with the base a

To find the nth order derivative of the base a logarithm, you need to express it in terms of the natural logarithm:
.
Applying formula (14), we find the nth derivative:
.

See also:

Complex derivatives. Logarithmic derivative.
The derivative of the 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 techniques and tricks for finding the derivative, in particular, with the logarithmic derivative.

Those readers with a low level of training should refer to the article How do I find the derivative? Examples of solutions, which will allow you to raise your skills almost from scratch. Next, you need to carefully study the page Derivative of a complex function, understand and solve all the examples I gave. This lesson is logically the third in a row, and after mastering it, you will confidently differentiate rather complex functions. It is undesirable to adhere to the position “Where else? And that's enough! ”, Because all examples and solutions are taken from real tests and are often found in practice.

Let's start with repetition. At the lesson Derivative of a complex function we have looked at a number of examples with detailed comments. In the course of studying differential calculus and other branches of mathematical analysis, you will have to differentiate very often, and it is not always convenient (and not always necessary) to write examples in great detail. Therefore, we will practice finding derivatives orally. 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 often not required, it is assumed that the student is able to find similar derivatives on the automatic autopilot. Imagine that at 3 am the phone rang, and a pleasant voice asked: "What is the derivative of the tangent of two Xs?" This should be followed by an almost instant 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 need to use only table of derivatives of elementary functions(if it is not remembered yet). If you have any difficulties, I recommend rereading the lesson. Derivative of a complex function.

, , ,
, , ,
, , ,

, , ,

, , ,

, , ,

, ,

Answers at the end of the lesson

Complex derivatives

After preliminary artillery preparation, examples with 3-4-5 function attachments will be less scary. Perhaps the following two examples will seem difficult to some, but if you understand them (someone will suffer), then almost everything else in the 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 the attachments. In cases where there are doubts, I recall a useful technique: we take the experimental value of "X", for example, and try (mentally or on a draft) to substitute this value in the "terrible expression".

1) First, we need to calculate the expression, which means that the amount is the deepest investment.

2) Then you need to calculate the logarithm:

4) Then raise the cosine to a cube:

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:

It seems without mistakes….

(1) 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 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 sound too difficult, but this is not yet 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 on the exam to check whether the student understands how to find the derivative of a complex function, or does not understand.

The next example is for a do-it-yourself solution.

Example 3

Find the derivative of a function

Hint: First, apply the linearity rules and the product differentiation rule

Complete solution and answer at the end of the tutorial.

Now is the time to move on to something more compact and cute.
It is not uncommon for an example to give a product of not two, but three functions. How to find the derivative of the product of three factors?

Example 4

Find the derivative of a function

First, let's see if it is possible to turn the product of three functions into the product of two functions? For example, if we had two polynomials in the product, then we could expand the brackets. But in this example, all functions are different: degree, exponent and logarithm.

In such cases, it is necessary consistently apply 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 a product of two factors and the rule does not work ?! There is nothing complicated:

Now it remains for the second time to apply the rule to the parenthesis:

You can still be perverted and put something outside the brackets, but in this case it is better to leave the answer in this form - it will be easier to check.

The considered 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.

Let's look at similar examples with fractions.

Example 6

Find the derivative of a function

There are several ways to go here:

Or like this:

But the solution will be written more compactly if, first of all, we use the rule for differentiating the quotient , taking for the entire numerator:

In principle, the example is solved, and if you leave it as it is, it will not be an error. But if you have time, it is always advisable to check on a draft, but is it possible to simplify the answer? Let us reduce 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 in finding the derivative, but in the case of banal school transformations. On the other hand, teachers often reject the assignment and ask to "bring 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 methods of finding the derivative, and now we will consider a typical case when the “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 differentiating a complex function:

But the very first step immediately plunges you into despondency - you have to take an unpleasant derivative from a fractional degree, and then also from a fraction.

That's why before how to take the derivative of the "fancy" logarithm, it is preliminarily simplified using the well-known school properties:

! If you have a practice notebook on hand, copy these formulas right there. If you don't have a notebook, redraw them on a piece of paper, as the rest of the lesson examples will revolve around these formulas.

The solution itself can be structured like this:

Let's transform the function:

Find the derivative:

Preconfiguring the function itself has greatly simplified the solution. Thus, when a similar logarithm is proposed for differentiation, it is always advisable to "break up" it.

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

We have seen similar examples recently. What to do? You can consistently apply the rule for differentiating the quotient, and then the rule for differentiating the work. 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 : since the function can take negative values, then, generally speaking, you need to use modules: that will disappear as a result of differentiation. However, the current design is also acceptable, where the defaults are taken into account complex values. But if with all the severity, then in both cases, a reservation should be made that.

Now you need to maximally "destroy" the logarithm of the right side (formulas in front of your eyes?). I will describe this process in great detail:

Actually, we proceed to differentiation.
We enclose both parts under the stroke:

The derivative of the right-hand side is quite simple, I will not comment on it, because if you are reading this text, you should confidently cope with it.

What about the left side?

On the left we have complex function... I foresee the question: "Why, there is also one letter" ygrek "under the logarithm?"

The fact is that this "one letter igrek" - ITSELF IS A FUNCTION(if not very clear, refer to the article Derived from an Implicit Function). Therefore, the logarithm is an external function, and the "game" is an internal function. And we use the rule of differentiating a complex function :

On the left side, as if by magic, we have a derivative. Further, according to the rule of proportion, we throw the "game" from the denominator of the left side to the top of the right side:

And now we recall what kind of “game” -function we discussed in differentiation? We look at the condition:

Final answer:

Example 12

Find the derivative of a function

This is an example for a do-it-yourself solution. A sample of the 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 the 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.

The derivative of the exponential function

We have not considered this function yet. An exponential function is a function in which and the degree and base depend on "x"... A classic example that will be given to you in any textbook or in 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-hand side, we got 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 the strokes:

Further actions are simple:

Finally:

If any transformation is not entirely clear, please carefully re-read the explanations in Example 11.

In practical tasks, the 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 embedded under the logarithm). When differentiating the constant, as we remember, it is better to immediately take out the sign of the derivative so that it does not get in the way under your feet; and of course we apply the familiar rule :

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:

1. Find the derivative of the function.
2. 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:

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

Solutions:

1. (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:

1. Find the derivatives of the functions and;
2. 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

1. 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:.
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 - 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:

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