When differentiating exponential power functions 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 a product is equal to the sum of logarithms, and 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 
Let
(1)
is a differentiable function of the variable x. First, we will consider it on the set of values x for which y takes positive values: . In the following, we will show that all the results obtained are also applicable for negative values of .
In some cases, in order to find the derivative of function (1), it is convenient to pre-logarithm it
,
and then calculate the derivative. Then, according to the rule of differentiation of a complex function,
.
From here
(2)
.
The derivative of the logarithm of a function is called the logarithmic derivative:
.
Logarithmic derivative of the function y = f(x) is the derivative of the natural logarithm of this function: (ln f(x))′.
The case of negative y values
Now consider the case when a variable can take both positive and negative values. In this case, take the logarithm of the modulus and find its derivative:
.
From here
(3)
.
That is, in the general case, you need to find the derivative of the logarithm of the modulus of the function.
Comparing (2) and (3) we have:
.
That is, the formal result of calculating the logarithmic derivative does not depend on whether we took the modulo or not. Therefore, when calculating the logarithmic derivative, we do not have to worry about what sign the function has.
This situation can be clarified using complex numbers. Let, for some values of x, be negative: . If we consider only real numbers, then the function is undefined. However, if we introduce complex numbers into consideration, we get the following:
.
That is, the functions and differ by a complex constant:
.
Since the derivative of a constant is zero, then
.
Property of the logarithmic derivative
From such a consideration it follows that the logarithmic derivative will not change if you multiply the function by an arbitrary constant :
.
Indeed, using properties of logarithm, formulas derivative sum And derivative of a constant, we have:
.
Application of logarithmic derivative
It is convenient to use the logarithmic derivative in cases where the original function consists of a product of power or exponential functions. In this case, the logarithm operation turns the product of functions into their sum. This simplifies the calculation of the derivative.
Example 1
Find the derivative of the function:
.
Solution
Let's logarithm the original function:
.
Let's differentiate with respect to the variable x.
In the table of derivatives we find:
.
We apply the rule of differentiation of complex functions.
;
;
;
;
(A1.1) .
Multiply by:
.
So, we found the logarithmic derivative:
.
From here we find the derivative of the original function:
.
Note
If we want to use only real numbers, then we should take the logarithm of the modulus of the original function:
.
Then
;
.
And we got formula (A1.1). Therefore the result has not changed.
Answer
Example 2
Using the logarithmic derivative, find the derivative of the function
.
Solution
Let's take logarithms:
(A2.1) .
Differentiate with respect to the variable x:
;
;
;
;
;
.
Multiply by:
.
From here we get the logarithmic derivative:
.
Derivative of the original function:
.
Note
Here the original function is non-negative: . It is defined at . If we do not assume that the logarithm can be defined for negative values of the argument, then formula (A2.1) should be written as follows:
.
Because the
And
,
this will not affect the final result.
Answer
Example 3
Find the derivative
.
Solution
We perform differentiation using the logarithmic derivative. Let's take a logarithm, taking into account that:
(A3.1) .
By differentiating, we obtain the logarithmic derivative.
;
;
;
(A3.2) .
Since then
.
Note
Let us carry out the calculations without the assumption that the logarithm can be defined for negative values of the argument. To do this, take the logarithm of the modulus of the original function:
.
Then instead of (A3.1) we have:
;
.
Comparing with (A3.2) we see that the result has not changed.
Lesson topic: “Differentiation of exponential and logarithmic functions. Antiderivative of the exponential function" in UNT assignments
Target : develop students’ skills in applying theoretical knowledge on the topic “Differentiation of exponential and logarithmic functions. Antiderivative of the exponential function" for solving UNT problems.
Tasks
Educational: systematize students’ theoretical knowledge, consolidate problem-solving skills on this topic.
Educational: develop memory, observation, logical thinking, mathematical speech of students, attention, self-esteem and self-control skills.
Educational: contribute:
developing a responsible attitude towards learning among students;
development of sustainable interest in mathematics;
creating positive internal motivation to study mathematics.
Teaching methods: verbal, visual, practical.
Forms of work: individual, frontal, in pairs.
During the classes
Epigraph: “The mind lies not only in knowledge, but also in the ability to apply knowledge in practice” Aristotle (slide 2)
I. Organizational moment.
II. Solving the crossword puzzle. (slide 3-21)
The 17th century French mathematician Pierre Fermat defined this line as “The straight line most closely adjacent to the curve in a small neighborhood of the point.”
Tangent
A function that is given by the formula y = log a x.
Logarithmic
A function that is given by the formula y = A X.
Indicative
In mathematics, this concept is used to find the speed of movement of a material point and the angular coefficient of a tangent to the graph of a function at a given point.
Derivative
What is the name of the function F(x) for the function f(x), if the condition F"(x) =f(x) is satisfied for any point from the interval I.
Antiderivative
What is the name of the relationship between X and Y, in which each element of X is associated with a single element of Y.
Derivative of displacement
Speed
A function that is given by the formula y = e x.
Exhibitor
If a function f(x) can be represented as f(x)=g(t(x)), then this function is called...
III. Mathematical dictation (slide 22)
1. Write down the formula for the derivative of the exponential function. ( A x)" = A x ln a
2. Write down the formula for the derivative of the exponential. (e x)" = e x
3. Write down the formula for the derivative of the natural logarithm. (ln x)"=
4. Write down the formula for the derivative of a logarithmic function. (log a x)"= 
5. Write down the general form of antiderivatives for the function f(x) = A X. F(x)= 
6. Write down the general form of antiderivatives for the function f(x) =, x≠0. F(x)=ln|x|+C
Check your work (answers on slide 23).
IV. Solving UNT problems (simulator)
A) No. 1,2,3,6,10,36 on the board and in the notebook (slide 24)
B) Work in pairs No. 19,28 (simulator) (slide 25-26)
V. 1. Find errors: (slide 27)
1) f(x)=5 e – 3х, f "(x)= – 3 e – 3х
2) f(x)=17 2x, f "(x)= 17 2x ln17
3) f(x)= log 5
(7x+1), f "(x)= 
4) f(x)= ln(9 – 4x), f "(x)= 
.
VI. Student presentation.
Epigraph: “Knowledge is such a precious thing that there is no shame in obtaining it from any source” Thomas Aquinas (slide 28)
VII. Homework No. 19,20 p.116
VIII. Test (reserve task) (slide 29-32)
IX. Lesson summary.
“If you want to participate in a big life, then fill your head with mathematics while you have the opportunity. She will then provide you with great help throughout your life” M. Kalinin (slide 33)
Differentiating exponential and logarithmic functions
1. Number e. Function y = e x, its properties, graph, differentiation
Let's consider an exponential function y=a x, where a > 1. For different bases a we get different graphs (Fig. 232-234), but you can notice that they all pass through the point (0; 1), they all have a horizontal asymptote y = 0 at , all of them are convexly facing downwards and, finally, they all have tangents at all their points. Let us draw, for example, a tangent to graphics function y=2x at point x = 0 (Fig. 232). If you make accurate constructions and measurements, you can make sure that this tangent forms an angle of 35° (approximately) with the x-axis.
Now let’s draw a tangent to the graph of the function y = 3 x, also at the point x = 0 (Fig. 233). Here the angle between the tangent and the x-axis will be greater - 48°. And for the exponential function y = 10 x in a similar
situation we get an angle of 66.5° (Fig. 234).
So, if the base a of the exponential function y=ax gradually increases from 2 to 10, then the angle between the tangent to the graph of the function at the point x=0 and the x-axis gradually increases from 35° to 66.5°. It is logical to assume that there is a base a for which the corresponding angle is 45°. This base must be enclosed between the numbers 2 and 3, since for the function y-2x the angle of interest to us is 35°, which is less than 45°, and for the function y=3 x it is equal to 48°, which is already a little more than 45 °. The base we are interested in is usually denoted by the letter e. It has been established that the number e is irrational, i.e. represents an infinite decimal non-periodic fraction:
e = 2.7182818284590...;
in practice it is usually assumed that e=2.7.
Comment(not very serious). It is clear that L.N. Tolstoy has nothing to do with the number e, however, in writing the number e, please note that the number 1828 is repeated twice in a row - the year of birth of L.N. Tolstoy.
The graph of the function y=e x is shown in Fig. 235. This is an exponential that differs from other exponentials (graphs of exponential functions with other bases) in that the angle between the tangent to the graph at point x=0 and the x-axis is 45°.
Properties of the function y = e x:
1)
2) is neither even nor odd;
3) increases;
4) not limited from above, limited from below;
5) has neither the largest nor the smallest values;
6) continuous;
7)
8) convex down;
9) differentiable.
Return to § 45, look at the list of properties of the exponential function y = a x for a > 1. You will find the same properties 1-8 (which is quite natural), and the ninth property associated with
we did not mention the differentiability of the function then. Let's discuss it now.
Let us derive a formula for finding the derivative y-ex. In this case, we will not use the usual algorithm, which we developed in § 32 and which has been successfully used more than once. In this algorithm, at the final stage it is necessary to calculate the limit, and our knowledge of the theory of limits is still very, very limited. Therefore, we will rely on geometric premises, considering, in particular, the very fact of the existence of a tangent to the graph of the exponential function beyond doubt (that is why we so confidently wrote down the ninth property in the above list of properties - the differentiability of the function y = e x).
1. Note that for the function y = f(x), where f(x) =ex, we already know the value of the derivative at the point x =0: f / = tan45°=1.
2. Let us introduce the function y=g(x), where g(x) -f(x-a), i.e. g(x)-ex" a. Fig. 236 shows the graph of the function y = g(x): it is obtained from the graph of the function y - fx) by shifting along the x axis by |a| scale units. Tangent to the graph of the function y = g (x) at point x-a is parallel to the tangent to the graph of the function y = f(x) at point x -0 (see Fig. 236), which means it forms an angle of 45° with the x axis. Using the geometric meaning of the derivative, we can write , that g(a) =tg45°;=1.
3. Let's return to the function y = f(x). We have:
4. We have established that for any value of a the relation is valid. Instead of the letter a, you can, of course, use the letter x; then we get
From this formula we obtain the corresponding integration formula:
A.G. Mordkovich Algebra 10th grade
Calendar-thematic planning in mathematics, video in mathematics online, Mathematics at school download
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