Derivative of function y lnx. Derivative of natural logarithm and logarithm to base a

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

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

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

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

Proof

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

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

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

.

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

And finally, apply property (6):
.
base logarithm e called natural logarithm. It is marked like this:
.
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 in 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 calculus and other areas 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 exponent.

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 . The inverse of the natural logarithm is the exponent:
.
Its derivative is determined by formula (9). Variables can be denoted by any letter. In formula (9), we replace the variable x with y:
.
Because , then
.
Then
.
The formula has been proven.

Now we prove the formula for the derivative of the natural logarithm using rules for differentiating a compound function. Since the functions and are inverse to each other, then
.
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 a complex function:
.
Here . Substitute into (10):
.
From here
.

Example

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

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 derivatives of ln 2x and ln 3x .

So, we are looking for the derivative of the function
y = log nx .
Let's represent 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 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 have substituted .

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 of the logarithm of the product:
.
- is a constant. Its derivative is zero. Then, according to the rule of differentiation of the sum, we have:
.

Answer

; ; .

Derivative of logarithm modulo x

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

Let's consider the case. Then the function looks like:
.
Its derivative is determined by formula (1):
.

Now consider the case . Then the function looks like:
,
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 to the base a, we have:
.

Higher order derivatives of the natural logarithm

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

Let's find the second order derivative:
.
Let's find the derivative of the third order:
.
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 mathematical induction.

Proof

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

Let us assume that formula (14) is satisfied for n = k . Let us prove that it follows from this that the formula is valid for n = k + 1 .

Indeed, for n = k we have:
.
Differentiate with respect to 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 nth order derivative, is valid for any n .

Higher-order derivatives of the logarithm to base a

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

Definition. Let the function \(y = f(x) \) be defined in some interval containing the point \(x_0 \) inside. Let's increment \(\Delta x \) to the argument so as not to leave this interval. Find the corresponding increment of the function \(\Delta y \) (when passing from the point \(x_0 \) to the point \(x_0 + \Delta x \)) and compose the relation \(\frac(\Delta y)(\Delta x) \). If there is a limit of this relation at \(\Delta x \rightarrow 0 \), then the indicated limit is called derivative function\(y=f(x) \) at the point \(x_0 \) and denote \(f"(x_0) \).

$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x_0) $$

The symbol y is often used to denote the derivative. Note that y" = f(x) is a new function, but naturally associated with the function y = f(x), defined at all points x at which the above limit exists . This function is called like this: derivative of the function y \u003d f (x).

The geometric meaning of the derivative consists of the following. If a tangent that is not parallel to the y axis can be drawn to the graph of the function y \u003d f (x) at a point with the abscissa x \u003d a, then f (a) expresses the slope of the tangent:
\(k = f"(a)\)

Since \(k = tg(a) \), the equality \(f"(a) = tg(a) \) is true.

And now we interpret the definition of the derivative in terms of approximate equalities. Let the function \(y = f(x) \) have a derivative at a particular point \(x \):
$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x) $$
This means that near the point x, the approximate equality \(\frac(\Delta y)(\Delta x) \approx f"(x) \), i.e. \(\Delta y \approx f"(x) \cdot\Deltax\). The meaningful meaning of the obtained approximate equality is as follows: the increment of the function is “almost proportional” to the increment of the argument, and the coefficient of proportionality is the value of the derivative at a given point x. For example, for the function \(y = x^2 \) the approximate equality \(\Delta y \approx 2x \cdot \Delta x \) is valid. If we carefully analyze the definition of the derivative, we will find that it contains an algorithm for finding it.

Let's formulate it.

How to find the derivative of the function y \u003d f (x) ?

1. Fix value \(x \), find \(f(x) \)
2. Increment \(x \) argument \(\Delta x \), move to a new point \(x+ \Delta x \), find \(f(x+ \Delta x) \)
3. Find the function increment: \(\Delta y = f(x + \Delta x) - f(x) \)
4. Compose the relation \(\frac(\Delta y)(\Delta x) \)
5. Calculate $$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) $$
This limit is the derivative of the function at x.

If the function y = f(x) has a derivative at the point x, then it is called differentiable at the point x. The procedure for finding the derivative of the function y \u003d f (x) is called differentiation functions y = f(x).

Let us discuss the following question: how are the continuity and differentiability of a function at a point related?

Let the function y = f(x) be differentiable at the point x. Then a tangent can be drawn to the graph of the function at the point M (x; f (x)) and, recall, the slope of the tangent is equal to f "(x). Such a graph cannot "break" at the point M, i.e., the function must be continuous at x.

It was reasoning "on the fingers". Let us present a more rigorous argument. If the function y = f(x) is differentiable at the point x, then the approximate equality \(\Delta y \approx f"(x) \cdot \Delta x \) holds. zero, then \(\Delta y \) will also tend to zero, and this is the condition for the continuity of the function at a point.

So, if a function is differentiable at a point x, then it is also continuous at that point.

The converse is not true. For example: function y = |x| is continuous everywhere, in particular at the point x = 0, but the tangent to the graph of the function at the “joint point” (0; 0) does not exist. If at some point it is impossible to draw a tangent to the function graph, then there is no derivative at this point.

One more example. The function \(y=\sqrt(x) \) is continuous on the entire number line, including at the point x = 0. And the tangent to the graph of the function exists at any point, including at the point x = 0. But at this point the tangent coincides with the y-axis, that is, it is perpendicular to the abscissa axis, its equation has the form x \u003d 0. There is no slope for such a straight line, which means that \ (f "(0) \) does not exist either

So, we got acquainted with a new property of a function - differentiability. How can you tell if a function is differentiable from the graph of a function?

The answer is actually given above. If at some point a tangent can be drawn to the graph of a function that is not perpendicular to the x-axis, then at this point the function is differentiable. If at some point the tangent to the graph of the function does not exist or it is perpendicular to the x-axis, then at this point the function is not differentiable.

Differentiation rules

The operation of finding the derivative is called differentiation. When performing this operation, you often have to work with quotients, sums, products of functions, as well as with "functions of functions", that is, complex functions. Based on the definition of the derivative, we can derive differentiation rules that facilitate this work. If C is a constant number and f=f(x), g=g(x) are some differentiable functions, then the following are true differentiation rules:

Table of derivatives of some functions

The operation of finding a derivative is called differentiation.

As a result of solving problems of finding derivatives of the simplest (and not very simple) functions by defining the derivative as the limit of the ratio of the increment to the increment of the argument, a table of derivatives and precisely defined rules of differentiation appeared. Isaac Newton (1643-1727) and Gottfried Wilhelm Leibniz (1646-1716) were the first to work in the field of finding derivatives.

Therefore, in our time, in order to find the derivative of any function, it is not necessary to calculate the above-mentioned limit of the ratio of the increment of the function to the increment of the argument, but only need to use the table of derivatives and the rules of differentiation. The following algorithm is suitable for finding the derivative.

To find the derivative, you need an expression under the stroke sign break down simple functions and determine what actions (product, sum, quotient) these functions are related. Further, we find the derivatives of elementary functions in the table of derivatives, and the formulas for the derivatives of the product, sum and quotient - in the rules of differentiation. The table of derivatives and differentiation rules are given after the first two examples.

Example 1 Find the derivative of a function

Solution. From the rules of differentiation we find out that the derivative of the sum of functions is the sum of derivatives of functions, i.e.

From the table of derivatives, we find out that the derivative of "X" is equal to one, and the derivative of the sine is cosine. We substitute these values ​​in the sum of derivatives and find the derivative required by the condition of the problem:

Example 2 Find the derivative of a function

Solution. Differentiate as a derivative of the sum, in which the second term with a constant factor, it can be taken out of the sign of the derivative:

If there are still questions about where something comes from, they, as a rule, become clear after reading the table of derivatives and the simplest rules of differentiation. We are going to them right now.

Table of derivatives of simple functions

1. Derivative of a constant (number). Any number (1, 2, 5, 200...) that is in the function expression. Always zero. This is very important to remember, as it is required very often
2. Derivative of the independent variable. Most often "x". Always equal to one. This is also important to remember
3. Derivative of degree. When solving problems, you need to convert non-square roots to a power.
4. Derivative of a variable to the power of -1
5. Derivative of the square root
6. Sine derivative
7. Cosine derivative
8. Tangent derivative
9. Derivative of cotangent
10. Derivative of the arcsine
11. Derivative of arc cosine
12. Derivative of arc tangent
13. Derivative of the inverse tangent
14. Derivative of natural logarithm
15. Derivative of a logarithmic function
16. Derivative of the exponent
17. Derivative of exponential function

Differentiation rules

1. Derivative of the sum or difference
2. Derivative of a product
2a. Derivative of an expression multiplied by a constant factor
3. Derivative of the quotient
4. Derivative of a complex function

Rule 1If functions

are differentiable at some point , then at the same point the functions

and

those. the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions.

Consequence. If two differentiable functions differ by a constant, then their derivatives are, i.e.

Rule 2If functions

are differentiable at some point , then their product is also differentiable at the same point

and

those. the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other.

Consequence 1. The constant factor can be taken out of the sign of the derivative:

Consequence 2. The derivative of the product of several differentiable functions is equal to the sum of the products of the derivative of each of the factors and all the others.

For example, for three multipliers:

Rule 3If functions

differentiable at some point and , then at this point their quotient is also differentiable.u/v , and

those. the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator.

Where to look on other pages

When finding the derivative of the product and the quotient in real problems, it is always necessary to apply several differentiation rules at once, so more examples on these derivatives are in the article."The derivative of a product and a quotient".

Comment. You should not confuse a constant (that is, a number) as a term in the sum and as a constant factor! In the case of a term, its derivative is equal to zero, and in the case of a constant factor, it is taken out of the sign of the derivatives. This is a typical mistake that occurs at the initial stage of studying derivatives, but as the average student solves several one-two-component examples, the average student no longer makes this mistake.

And if, when differentiating a product or a quotient, you have a term u"v, wherein u- a number, for example, 2 or 5, that is, a constant, then the derivative of this number will be equal to zero and, therefore, the entire term will be equal to zero (such a case is analyzed in example 10).

Another common mistake is the mechanical solution of the derivative of a complex function as the derivative of a simple function. That's why derivative of a complex function devoted to a separate article. But first we will learn to find derivatives of simple functions.

Along the way, you can not do without transformations of expressions. To do this, you may need to open in new windows manuals Actions with powers and roots and Actions with fractions .

If you are looking for solutions to derivatives with powers and roots, that is, when the function looks like , then follow the lesson " Derivative of the sum of fractions with powers and roots".

If you have a task like , then you are in the lesson "Derivatives of simple trigonometric functions".

Step by step examples - how to find the derivative

Example 3 Find the derivative of a function

Solution. We determine the parts of the function expression: the entire expression represents the product, and its factors are sums, in the second of which one of the terms contains a constant factor. We apply the product differentiation rule: the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other:

Next, we apply the rule of differentiation of the sum: the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions. In our case, in each sum, the second term with a minus sign. In each sum, we see both an independent variable, the derivative of which is equal to one, and a constant (number), the derivative of which is equal to zero. So, "x" turns into one, and minus 5 - into zero. In the second expression, "x" is multiplied by 2, so we multiply two by the same unit as the derivative of "x". We get the following values ​​of derivatives:

We substitute the found derivatives into the sum of products and obtain the derivative of the entire function required by the condition of the problem:

Example 4 Find the derivative of a function

Solution. We are required to find the derivative of the quotient. We apply the formula for differentiating a quotient: the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator. We get:

We have already found the derivative of the factors in the numerator in Example 2. Let's also not forget that the product, which is the second factor in the numerator in the current example, is taken with a minus sign:

If you are looking for solutions to such problems in which you need to find the derivative of a function, where there is a continuous pile of roots and degrees, such as, for example, then welcome to class "The derivative of the sum of fractions with powers and roots" .

If you need to learn more about the derivatives of sines, cosines, tangents and other trigonometric functions, that is, when the function looks like , then you have a lesson "Derivatives of simple trigonometric functions" .

Example 5 Find the derivative of a function

Solution. In this function, we see a product, one of the factors of which is the square root of the independent variable, with the derivative of which we familiarized ourselves in the table of derivatives. According to the product differentiation rule and the tabular value of the derivative of the square root, we get:

Example 6 Find the derivative of a function

Solution. In this function, we see the quotient, the dividend of which is the square root of the independent variable. According to the rule of differentiation of the quotient, which we repeated and applied in example 4, and the tabular value of the derivative of the square root, we get:

To get rid of the fraction in the numerator, multiply the numerator and denominator by .