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Derivative natural logarithm of x is equal to one divided by x:
(1)
(ln x)′ =.
The derivative of the logarithm to base a is equal to one divided by the variable x multiplied by the natural logarithm of a:
(2)
(log a x)′ =.
Proof
Let there be some positive number not equal to one. Consider a function depending on a variable x, which is a logarithm to the base:
.
This function is defined at .
(3)
.
Let's find its derivative with respect to the variable x.
By definition, the derivative is the following limit: Let's transform this expression to reduce it to known mathematical properties and rules. To do this we need to know the following facts:
(4)
;
(5)
;
(6)
;
A) Properties of the logarithm. We will need the following formulas:
(7)
.
B)
Continuity of the logarithm and the property of limits for a continuous function: Here is a function that has a limit and this limit is positive.
(8)
.
IN)
.
The meaning of the second remarkable limit:
.
Let's apply these facts to our limit. First we transform the algebraic expression
.
To do 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 to base e called
.
natural logarithm
.
. It is designated as follows:
Then ;
Thus, we obtained formula (2) for the derivative of the logarithm.
.
Derivative of the natural logarithm
(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 through the natural logarithm using property (6):
.
The derivative of the logarithm with respect to the base can be found from formula (1), if you take the constant out of the differentiation sign:
.
Other ways to prove the derivative of a logarithm
Here we assume that we know the formula for the derivative of the exponential:
(9)
.
Then we can derive the formula for the derivative of the natural logarithm, given that the logarithm is the inverse function of the exponential.
Let us prove the formula for the derivative of the natural logarithm, applying the formula for the derivative of the inverse function:
.
In our case . Inverse function the exponential to the natural logarithm is:
.
Its derivative is determined by formula (9). Variables can be designated by 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 differentiation rules complex function
. Since the functions and are inverse to each other, then
.
Let's differentiate this equation with respect to the variable x:
(10)
.
The derivative of x is equal to one:
.
We apply the rule of differentiation of complex functions:
.
Here . Let's substitute in (10):
.
From here
.
Example
Find derivatives of ln 2x, ln 3x And lnnx.
Solution
The original functions have a similar form. Therefore we will find the derivative of the function y = log nx. Then we substitute 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 = log nx
.
Let's imagine this function as a complex function consisting of two functions:
1)
Functions depending on a variable: ;
2)
Functions depending on a variable: .
Then the original function is composed of the functions and :
.
Let's 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 it up.
So we found:
(11)
.
We see that the derivative does not depend on n.
.
This result is quite natural if we transform the original function using the formula for the logarithm of the product:
.
- this is a constant. Its derivative is zero. Then, according to the rule of differentiation of the sum, we have:
; ; .
Answer
Derivative of the logarithm of modulus x Let's find the derivative of another very important function
(12)
.
- natural logarithm of modulus x:
.
Let's consider the case.
.
Now let's consider the case.
,
Then the function looks like:
Where .
.
Then
.
But we also found the derivative of this function in the example above. It does not depend on n and is equal to
.
We combine these two cases into one formula:
.
Accordingly, for the logarithm to base a, we have:
Derivatives of higher orders of the natural logarithm
.
Consider the function
(13)
.
We found its first-order derivative:
.
Let's find the second order derivative:
.
Let's find the third order derivative:
.
Let's find the fourth order derivative:
(14)
.
You can notice that the nth order derivative has the form:
Proof
Let us prove this by mathematical induction.
.
Let us substitute the value n = 1 into formula (14): 1
Since , then when n =
, formula (14) is valid. + 1 .
Let us assume that formula (14) is satisfied for n = k.
.
Let us prove that this implies that the formula is valid for n = k
.
Indeed, for n = k we have:
.
Differentiate with respect to the variable x: 1
So we got: 1
.
This formula coincides with formula (14) for n = k +
.
Thus, from the assumption that formula (14) is valid for n = k, it follows that formula (14) is valid for n = k +
.
Therefore, formula (14), for the nth order derivative, is valid for any n.
.
Derivatives of higher orders of the logarithm to base a
To find the nth order derivative of a logarithm to base a, you need to express it in terms of the natural logarithm:
Applying formula (14), we find the nth derivative:
Do you feel like there is still a lot of time before the exam? Is this a month? Two? Year? Practice shows that a student copes best with an exam if he begins to prepare for it in advance. There are many difficult tasks in the Unified State Exam that stand in the way of schoolchildren and future applicants to the highest scores. You need to learn to overcome these obstacles, and besides, it’s not difficult to do. You need to understand the principle of working with various tasks from tickets. Then there will be no problems with the new ones.
Now let's look at the second concept. The derivative of a function in any form is a concept that characterizes the change in a function at a given point. However, this school program, and if you have problems with these concepts individually, it is worth repeating the topic.
Derivative of logarithm
IN Unified State Exam assignments On this topic, several problems can be given as examples. To begin with, the simplest logarithmic derivative. It is necessary to find the derivative of the following function.
We need to find the next derivative
There is a special formula.
In this case x=u, log3x=v. We substitute the values from our function into the formula.
The derivative of x will be equal to one. The logarithm is a little more difficult. But you will understand the principle if you simply substitute the values. Recall that the derivative of lg x is the derivative of the decimal logarithm, and the derivative of ln x is the derivative of the natural logarithm (based on e).
Now simply plug the resulting values into the formula. Try it yourself, then we’ll check the answer.
What could be the problem here for some? We introduced the concept of natural logarithm. Let's talk about it, and at the same time figure out how to solve problems with it. You won’t see anything complicated, especially when you understand the principle of its operation. You should get used to it, since it is often used in mathematics (in higher educational institutions especially).
Derivative of the natural logarithm
At its core, it is the derivative of the logarithm to base e (this is irrational number, which is approximately 2.7). In fact, ln is very simple, so it is often used in mathematics in general. Actually, solving the problem with it will not be a problem either. It is worth remembering that the derivative of the natural logarithm to the base e will be equal to one divided by x. The solution to the following example will be the most revealing.
Let's imagine it as a complex function consisting of two simple ones.
It is enough to convert
We are looking for the derivative of u with respect to x
When differentiating, it is indicative power function or cumbersome fractional expressions, it is convenient to use the logarithmic derivative. In this article we will look at examples of its application with detailed solutions.
Further presentation assumes the ability to use the table of derivatives, differentiation rules and knowledge of the formula for the derivative of a complex function.
Derivation of the formula for the logarithmic derivative.
First, we take logarithms to the base e, simplify the form of the function using the properties of the logarithm, and then find the derivative of the implicitly specified function: 
For example, let's find the derivative of an exponential power function x to the power x.
Taking logarithms gives . According to the properties of the logarithm. Differentiating both sides of the equality leads to the result: 
Answer: 
.
The same example can be solved without using the logarithmic derivative. You can carry out some transformations and move from differentiating an exponential power function to finding the derivative of a complex function: 
Example.
Find the derivative of a function 
.
Solution.
In this example the function 
is a fraction and its derivative can be found using the rules of differentiation. But due to the cumbersomeness of the expression, this will require many transformations. In such cases, it is more reasonable to use the logarithmic derivative formula 
. Why? You will understand now.
Let's find it first. In transformations we will use the properties of the logarithm (the logarithm of a fraction is equal to the difference of logarithms, and the logarithm of the product equal to the sum logarithms, and also the degree of the expression under the logarithm sign can be taken out as a coefficient in front of the logarithm): 
These transformations led us to a fairly simple expression, the derivative of which is easy to find: 
We substitute the result obtained into the formula for the logarithmic derivative and get the answer:
To consolidate the material, we will give a couple more examples without detailed explanations.
Example.
Find the derivative of an exponential power function 
