What is the derivative of e? Typical errors when calculating the derivative
How to find the derivative, how to take the derivative? On this lesson we will learn to find derivatives of functions. But before studying this page, I strongly recommend that you familiarize yourself with the methodological materialHot formulas school course mathematicians. The reference manual can be opened or downloaded on the page Mathematical formulas and tables . Also from there we will needDerivatives table, it is better to print it out; you will often have to refer to it, not only now, but also offline.
Eat? Let's get started. I have two news for you: good and very good. The good news is this: to learn how to find derivatives, you don't need to know or understand what a derivative is. Moreover, it is more appropriate to digest the definition of the derivative of a function, the mathematical, physical, geometric meaning of the derivative later, since a high-quality study of the theory, in my opinion, requires the study of a number of other topics, as well as some practical experience.
And now our task is to master these same derivatives technically. The very good news is that learning to take derivatives is not that difficult; there is a fairly clear algorithm for solving (and explaining) this task; integrals or limits, for example, are more difficult to master.
I advise you to study the topic in the following order: first, This article. Then you need to read the most important lesson Derivative complex function. These two basic classes will take your skills from scratch. Next you can get acquainted with more complex derivatives in the article Complex derivatives.
Logarithmic derivative. If the bar is too high, read the thing first Protozoa typical tasks with derivative. In addition to the new material, the lesson covers other, simpler types of derivatives, and is a great opportunity to improve your differentiation technique. In addition, in tests Almost always there are tasks to find derivatives of functions that are specified implicitly or parametrically. There is also such a lesson: Derivatives of implicit and parametrically defined functions.
I will try in an accessible form, step by step, to teach you how to find derivatives of functions. All information is presented in detail, in simple words.
Actually, let's look at an example right away: Example 1
Find the derivative of the function Solution: 
This simplest example, please find it in the derivatives table elementary functions. Now let's look at the solution and analyze what happened? And the following thing happened:
we had a function that, as a result of the solution, turned into a function.
To put it quite simply, to find the derivative
function, you need to turn it into another function according to certain rules . Look again at the table of derivatives - there functions turn into other functions. The only one
the exception is the exponential function, which
turns into itself. The operation of finding the derivative is calleddifferentiation.
Notation: The derivative is denoted by or.
ATTENTION, IMPORTANT! Forgetting to put a stroke (where it is necessary), or to draw an extra stroke (where it is not necessary) is a GROSS MISTAKE! A function and its derivative are two different functions!
Let's return to our table of derivatives. From this table it is desirable memorize: rules of differentiation and derivatives of some elementary functions, especially:
derivative of the constant:
Where is a constant number; derivative power function:
In particular: , , .
Why remember? This knowledge is basic knowledge about derivatives. And if you cannot answer the teacher’s question “What is the derivative of a number?”, then your studies at the university may end for you (I personally know two real cases from life). In addition, these are the most common formulas that we have to use almost every time we come across derivatives.
IN In reality, simple tabular examples are rare; usually, when finding derivatives, differentiation rules are first used, and then a table of derivatives of elementary functions.
IN of this connection we proceed to considerdifferentiation rules:
1) A constant number can (and should) be taken out of the derivative sign
Where is a constant number (constant) Example 2
Find the derivative of a function
Let's look at the table of derivatives. The derivative of the cosine is there, but we have .
It's time to use the rule, we take the constant factor out of the sign of the derivative:
Now we convert our cosine according to the table:
Well, it’s advisable to “comb” the result a little - put the minus sign in first place, at the same time getting rid of the brackets:
2) The derivative of the sum is equal to the sum of the derivatives
Find the derivative of a function
Let's decide. As you probably already noticed, the first step that is always performed when finding a derivative is that we enclose the entire expression in parentheses and put a prime at the top right:
Let's apply the second rule:
Please note that for differentiation, all roots and powers must be represented in the form , and if they are in the denominator, then
move them up. How to do this is discussed in my teaching materials.
Now let’s remember the first rule of differentiation - we take the constant factors (numbers) outside the sign of the derivative:
Usually, during the solution, these two rules are applied simultaneously (so as not to rewrite a long expression again).
All functions located under the strokes are elementary table functions; using the table we carry out the transformation:
You can leave everything as is, since there are no more strokes, and the derivative has been found. However, expressions like this usually simplify:
It is advisable to represent all powers of the form again in the form of roots,
powers with negative exponents – discard into the denominator. Although you don't have to do this, it won't be a mistake.
Find the derivative of a function 
Try to solve this example yourself (answer at the end of the lesson).
3) Derivative of the product of functions
It seems that the analogy suggests the formula ...., but the surprise is that:
This is an unusual rule(as, in fact, others) follows from derivative definitions. But we’ll hold off on the theory for now – now it’s more important to learn how to solve:
Find the derivative of a function
Here we have the product of two functions depending on . First we apply our strange rule, and then we transform the functions using the derivative table:
Difficult? Not at all, quite accessible even for a teapot.
Find the derivative of a function
This function contains the sum and product of two functions - quadratic trinomial and logarithm. From school we remember that multiplication and division take precedence over addition and subtraction.
It's the same here. FIRST we use the product differentiation rule:
Now for the bracket we use the first two rules:
As a result of applying the rules of differentiation under the strokes, we are left with only elementary functions; using the table of derivatives, we transform them into other functions:
With some experience in finding derivatives, simple derivatives do not seem to need to be described in such detail. In general, they are usually decided orally, and it is immediately written down that 
.
Find the derivative of a function 
This is an example for independent decision(answer at the end of the lesson)
4) Derivative of quotient functions
A hatch opened in the ceiling, don't be alarmed, it's a glitch. But this is the harsh reality:
Find the derivative of a function
What’s missing here – sum, difference, product, fraction…. Where to start?! There are doubts, there are no doubts, but, IN ANY CASE, first we draw brackets and put a stroke at the top right:
Now we look at the expression in brackets, how can we simplify it? In this case, we notice a factor, which, according to the first rule, it is advisable to take out the sign of the derivative:
At the same time, we get rid of the parentheses in the numerator, which are no longer needed. Generally speaking, constant factors when finding the derivative
You may not have to endure them, but in this case they will “get under your feet,” which clutters up and complicates the solution.
Let's look at our expression in brackets. We have addition, subtraction and division. From school we remember that division is done first. And here - first we apply the rule of differentiation of quotients:
Thus, our terrible derivative has been reduced to derivatives of two simple expressions. We apply the first and second rules, here we will do it orally, I hope you have already become a little familiar with derivatives:
There are no more strokes, the task is completed.
In practice, the answer is usually (but not always) simplified using “school” methods:
Find the derivative of a function
This is an example for you to solve on your own (answer at the end of the lesson). From time to time there are tricky puzzles:
Find the derivative of a function 
Let's look at this function. Here's a fraction again. However, before using the rule for differentiating quotients (and it can be used), it always makes sense to see if it is possible to simplify the fraction itself, or get rid of it altogether?
The point is that the formula 
It’s quite cumbersome, and I don’t want to use it at all.
In this case, you can divide the numerator by the denominator term by term. Let's transform the function:
Well, that’s a completely different matter, now differentiation is simple and pleasant:
Find the derivative of a function
Here the situation is similar, let’s turn our fraction into a product, to do this we raise the exponent to the numerator, changing the sign of the exponent:
It’s still easier to differentiate the product:
Find the derivative of a function This is an example for solving it yourself (answer at the end of the lesson).
5) Derivative of a complex function
This rule also occurs very often. But you can tell a lot about it, so I created a separate lesson on the topic Derivative of a complex function.
I wish you success!
Example 4: 
. During the decision
In this example, you should pay attention to the fact that and are constant numbers, it doesn’t matter what they are equal to, it is important that they are constants. Therefore, it is taken out of the derivative sign, and .
Example 7: 
Example 9: 
Date: 05/10/2015
How to find the derivative?
Rules of differentiation.
To find the derivative of any function, you need to master only three concepts:
2. Rules of differentiation.
3. Derivative of a complex function.
Exactly in that order. This is a hint.)
Of course, it would be nice to have an idea about derivatives in general). What a derivative is and how to work with the table of derivatives is clearly explained in the previous lesson. Here we will deal with the rules of differentiation.
Differentiation is the operation of finding the derivative. There is nothing more hidden behind this term. Those. expressions "find the derivative of a function" And "differentiate a function"- it's the same thing.
Expression "rules of differentiation" refers to finding the derivative from arithmetic operations. This understanding helps a lot to avoid confusion in your head.
Let's concentrate and remember all, all, all arithmetic operations. There are four of them). Addition (sum), subtraction (difference), multiplication (product), and division (quotient). Here they are, the rules of differentiation:
The plate shows five rules on four arithmetic operations. I didn’t get shortchanged.) It’s just that rule 4 is an elementary consequence of rule 3. But it is so popular that it makes sense to write (and remember!) it as an independent formula.
Under the designations U And V some (absolutely any!) functions are implied U(x) And V(x).
Let's look at a few examples. First - the simplest ones.
Find the derivative of the function y=sinx - x 2
Here we have difference two elementary functions. We apply rule 2. We will assume that sinx is a function U, and x 2 is the function V. We have every right write:
y" = (sinx - x 2)" = (sinx)"- (x 2)"
That’s better, right?) All that remains is to find the derivatives of sine and square of x. There is a table of derivatives for this purpose. We just look for the functions we need in the table ( sinx And x 2), look at what derivatives they have and write down the answer:
y" = (sinx)" - (x 2)" = cosx - 2x
That's it. Rule 1 of sum differentiation works exactly the same.
What if we have several terms? No problem.) We break the function into terms and look for the derivative of each term independently of the others. For example:
Find the derivative of the function y=sinx - x 2 +cosx - x +3
We boldly write:
y" = (sinx)" - (x 2)" + (cosx)" - (x)" + (3)"
At the end of the lesson I will give tips to make life easier when differentiating.)
1. Before differentiation, see if it is possible to simplify the original function.
2. In complicated examples, we describe the solution in detail, with all the parentheses and dashes.
3. When differentiating fractions with a constant number in the denominator, we turn division into multiplication and use rule 4.
The process of finding the derivative of a function is called differentiation. The derivative has to be found in a number of problems in the course mathematical analysis. For example, when finding extremum points and inflection points of a function graph.
How to find?
To find the derivative of a function you need to know the table of derivatives of elementary functions and apply the basic rules of differentiation:
- Moving the constant beyond the sign of the derivative: $$ (Cu)" = C(u)" $$
- Derivative of the sum/difference of functions: $$ (u \pm v)" = (u)" \pm (v)" $$
- Derivative of the product of two functions: $$ (u \cdot v)" = u"v + uv" $$
- Derivative of a fraction: $$ \bigg (\frac(u)(v) \bigg)" = \frac(u"v - uv"))(v^2) $$
- Derivative of a complex function: $$ (f(g(x)))" = f"(g(x)) \cdot g"(x) $$
Examples of solutions
| Example 1 |
| Find the derivative of the function $ y = x^3 - 2x^2 + 7x - 1 $ |
| Solution |
|
The derivative of the sum/difference of functions is equal to the sum/difference of derivatives: $$ y" = (x^3 - 2x^2 + 7x - 1)" = (x^3)" - (2x^2)" + (7x)" - (1)" = $$ Using the rule for the derivative of a power function $ (x^p)" = px^(p-1) $ we have: $$ y" = 3x^(3-1) - 2 \cdot 2 x^(2-1) + 7 - 0 = 3x^2 - 4x + 7 $$ It was also taken into account that the derivative of a constant is equal to zero. If you cannot solve your problem, then send it to us. We will provide detailed solution. You will be able to view the progress of the calculation and gain information. This will help you get your grade from your teacher in a timely manner! |
| Answer |
| $$y" = 3x^2 - 4x + 7 $$ |
Solving physical problems or examples in mathematics is completely impossible without knowledge of the derivative and methods for calculating it. The derivative is one of the most important concepts in 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 derivative
Let there be a function f(x) , specified in a certain interval (a, b) . Points x and x0 belong to this interval. When x changes, the function itself changes. Changing the argument - the difference in its values x-x0 . This difference is written as delta x and is called argument increment. A change or increment of a function is the difference between the values of a function at two points. Definition of derivative:
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's the point of finding such a limit? And here's what it is:
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.
Physical meaning of the derivative: the derivative of the path with respect to time is equal to the speed of rectilinear motion.
Indeed, since school days everyone knows that speed is a particular path x=f(t) and time t . Average speed for a certain period of time:
To find out the speed of movement at a moment in time t0 you need to calculate the limit:
Rule one: set a constant
The constant can be taken out of the derivative sign. Moreover, it needs to be done. When solving examples in mathematics, take it as a rule - If you can simplify an expression, be sure to simplify it .
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 the function:
Rule three: 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:
Solution: 
It is important to talk about calculating derivatives of complex functions here. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable.
In the above example we come across 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 calculate 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: derivative of the quotient of two functions
Formula for determining the derivative of the 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 questions on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult test and understand the tasks, even if you have never done derivative calculations before.
