It is absolutely impossible to solve physical problems or examples in mathematics without knowledge about the derivative and methods for calculating it. The derivative is one of the most important concepts of 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 the derivative
Let there be a function f(x) , given in some interval (a,b) . The points x and x0 belong to this interval. When x changes, the function itself changes. Argument change - difference of its values x-x0 . This difference is written as delta x and is called argument increment. The change or increment of a function is the difference between the values of the function at two points. Derivative definition:
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 is the point in finding such a limit? But which one:
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 derivative: the time derivative of the path is equal to the speed of the rectilinear motion.
Indeed, since school days, everyone knows that speed is a private path. x=f(t) and time t . average speed for some period of time:
To find out the speed of movement at a time t0 you need to calculate the limit:
Rule one: take out the constant
The constant can be taken out of the sign of the derivative. Moreover, it must be done. When solving examples in mathematics, take as a rule - if you can simplify the expression, be sure to simplify .
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 a function:
Rule three: the 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:
Decision: 
Here it is important to say about the calculation of derivatives of complex functions. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument by the derivative of the intermediate argument with respect to the independent variable.
In the above example, we encounter 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 consider 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: The derivative of the quotient of two functions
Formula for determining the derivative of a quotient of two functions:
We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it sounds, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.
With any question on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult control and deal with tasks, even if you have never dealt with the calculation of derivatives before.
Here is a summary table for convenience and clarity when studying the topic.
|
Constanty=C Power function y = x p (x p)" = p x p - 1 |
Exponential functiony = x (a x)" = a x ln a In particular, whena = ewe have y = e x (e x)" = e x |
|
logarithmic function (log a x) " = 1 x ln a In particular, whena = ewe have y = log x (ln x)" = 1 x |
Trigonometric functions (sin x) "= cos x (cos x)" = - sin x (t g x) " = 1 cos 2 x (c t g x)" = - 1 sin 2 x |
|
Inverse trigonometric functions (a r c sin x) " = 1 1 - x 2 (a r c cos x) " = - 1 1 - x 2 (a r c t g x) " = 1 1 + x 2 (a r c c t g x) " = - 1 1 + x 2 |
Hyperbolic functions (s h x) " = c h x (c h x) " = s h x (t h x) " = 1 c h 2 x (c t h x) " = - 1 s h 2 x |
Let us analyze how the formulas of the specified table were obtained, or, in other words, we will prove the derivation of formulas for derivatives for each type of function.
Derivative of a constant
Proof 1In order to bring out this formula, we take as a basis the definition of the derivative of a function at a point. We use x 0 = x, where x takes on the value of any real number, or, in other words, x is any number from the domain of the function f (x) = C . Let's write the limit of the ratio of the increment of the function to the increment of the argument as ∆ x → 0:
lim ∆ x → 0 ∆ f (x) ∆ x = lim ∆ x → 0 C - C ∆ x = lim ∆ x → 0 0 ∆ x = 0
Please note that the expression 0 ∆ x falls under the limit sign. It is not the uncertainty of “zero divided by zero”, since the numerator contains not an infinitesimal value, but zero. In other words, the increment of a constant function is always zero.
So, the derivative of the constant function f (x) = C is equal to zero over the entire domain of definition.
Example 1
Given constant functions:
f 1 (x) = 3 , f 2 (x) = a , a ∈ R , f 3 (x) = 4 . 13 7 22 , f 4 (x) = 0 , f 5 (x) = - 8 7
Decision
Let us describe the given conditions. In the first function we see the derivative of the natural number 3 . In the following example, you need to take the derivative of a, where a- any real number. The third example gives us the derivative of the irrational number 4 . 13 7 22 , the fourth - the derivative of zero (zero is an integer). Finally, in the fifth case we have the derivative rational fraction - 8 7 .
Answer: derivatives set functions is zero for any real x(over the entire domain of definition)
f 1 " (x) = (3) " = 0 , f 2 " (x) = (a) " = 0 , a ∈ R , f 3 " (x) = 4 . 13 7 22 " = 0 , f 4 " (x) = 0 " = 0 , f 5 " (x) = - 8 7 " = 0
Power function derivative
Let's move on to power function and the formula of its derivative, which has the form: (x p) " = p x p - 1, where the exponent p is any real number.
Proof 2
We present the proof of the formula when the exponent is natural number: p = 1 , 2 , 3 , …
Again, we rely on the definition of a derivative. Let's write the limit of the ratio of the increment of the power function to the increment of the argument:
(x p) " = lim ∆ x → 0 = ∆ (x p) ∆ x = lim ∆ x → 0 (x + ∆ x) p - x p ∆ x
To simplify the expression in the numerator, we use Newton's binomial formula:
(x + ∆ x) p - x p = C p 0 + x p + C p 1 x p - 1 ∆ x + C p 2 x p - 2 (∆ x) 2 + . . . + + C p p - 1 x (∆ x) p - 1 + C p p (∆ x) p - x p = = C p 1 x p - 1 ∆ x + C p 2 x p - 2 (∆ x) 2 + . . . + C p p - 1 x (∆ x) p - 1 + C p p (∆ x) p
Thus:
(x p) " = lim ∆ x → 0 ∆ (x p) ∆ x = lim ∆ x → 0 (x + ∆ x) p - x p ∆ x = = lim ∆ x → 0 (C p 1 x p - 1 ∆ x + C p 2 x p - 2 (∆ x) 2 + . . . + C p p - 1 x (∆ x) p - 1 + C p p (∆ x) p) ∆ x = = lim ∆ x → 0 (C p 1 x p - 1 + C p 2 x p - 2 ∆ x + . . . + C p p - 1 x (∆ x) p - 2 + C p p (∆ x) p - 1) = = C p 1 x p - 1 + 0 + 0 + . . . + 0 = p! 1! (p - 1)! x p - 1 = p x p - 1
So, we proved the formula for the derivative of a power function when the exponent is a natural number.
Proof 3
To give proof for the case when p- any real number other than zero, we use the logarithmic derivative (here we should understand the difference from the derivative of the logarithmic function). To have a more complete understanding, it is desirable to study the derivative of the logarithmic function and additionally deal with the derivative of an implicitly given function and the derivative of a complex function.
Consider two cases: when x positive and when x are negative.
So x > 0 . Then: x p > 0 . We take the logarithm of the equality y \u003d x p to the base e and apply the property of the logarithm:
y = x p ln y = ln x p ln y = p ln x
At this stage, an implicitly defined function has been obtained. Let's define its derivative:
(ln y) " = (p ln x) 1 y y " = p 1 x ⇒ y " = p y x = p x p x = p x p - 1
Now we consider the case when x- a negative number.
If the indicator p there is even number, then the power function is also defined for x< 0 , причем является четной: y (x) = - y ((- x) p) " = - p · (- x) p - 1 · (- x) " = = p · (- x) p - 1 = p · x p - 1
Then xp< 0 и возможно составить доказательство, используя логарифмическую производную.
If a p is an odd number, then the power function is defined for x< 0 , причем является нечетной: y (x) = - y (- x) = - (- x) p . Тогда x p < 0 , а значит логарифмическую производную задействовать нельзя. В такой ситуации возможно взять за основу доказательства правила дифференцирования и правило нахождения производной сложной функции:
y "(x) \u003d (- (- x) p) " \u003d - ((- x) p) " \u003d - p (- x) p - 1 (- x) " = \u003d p (- x) p - 1 = p x p - 1
The last transition is possible because if p is an odd number, then p - 1 either an even number or zero (for p = 1), therefore, for negative x the equality (- x) p - 1 = x p - 1 is true.
So, we have proved the formula for the derivative of a power function for any real p.
Example 2
Given functions:
f 1 (x) = 1 x 2 3 , f 2 (x) = x 2 - 1 4 , f 3 (x) = 1 x log 7 12
Determine their derivatives.
Decision
We transform part of the given functions into a tabular form y = x p , based on the properties of the degree, and then use the formula:
f 1 (x) \u003d 1 x 2 3 \u003d x - 2 3 ⇒ f 1 "(x) \u003d - 2 3 x - 2 3 - 1 \u003d - 2 3 x - 5 3 f 2 "(x) \u003d x 2 - 1 4 = 2 - 1 4 x 2 - 1 4 - 1 = 2 - 1 4 x 2 - 5 4 f 3 (x) = 1 x log 7 12 = x - log 7 12 ⇒ f 3 "( x) = - log 7 12 x - log 7 12 - 1 = - log 7 12 x - log 7 12 - log 7 7 = - log 7 12 x - log 7 84
Derivative of exponential function
Proof 4We derive the formula for the derivative, based on the definition:
(a x) " = lim ∆ x → 0 a x + ∆ x - a x ∆ x = lim ∆ x → 0 a x (a ∆ x - 1) ∆ x = a x lim ∆ x → 0 a ∆ x - 1 ∆ x = 0 0
We got uncertainty. To expand it, we write a new variable z = a ∆ x - 1 (z → 0 as ∆ x → 0). In this case a ∆ x = z + 1 ⇒ ∆ x = log a (z + 1) = ln (z + 1) ln a . For the last transition, the formula for the transition to a new base of the logarithm is used.
Let's perform a substitution in the original limit:
(a x) " = a x lim ∆ x → 0 a ∆ x - 1 ∆ x = a x ln a lim ∆ x → 0 1 1 z ln (z + 1) = = a x ln a lim ∆ x → 0 1 ln (z + 1) 1 z = a x ln a 1 ln lim ∆ x → 0 (z + 1) 1 z
Recall the second remarkable limit and then we obtain the formula for the derivative exponential function:
(a x) " = a x ln a 1 ln lim z → 0 (z + 1) 1 z = a x ln a 1 ln e = a x ln a
Example 3
The exponential functions are given:
f 1 (x) = 2 3 x , f 2 (x) = 5 3 x , f 3 (x) = 1 (e) x
We need to find their derivatives.
Decision
We use the formula for the derivative of the exponential function and the properties of the logarithm:
f 1 "(x) = 2 3 x" = 2 3 x ln 2 3 = 2 3 x (ln 2 - ln 3) f 2 "(x) = 5 3 x" = 5 3 x ln 5 1 3 = 1 3 5 3 x ln 5 f 3 "(x) = 1 (e) x" = 1 e x " = 1 e x ln 1 e = 1 e x ln e - 1 = - 1 e x
Derivative of a logarithmic function
Proof 5We present the proof of the formula for the derivative of the logarithmic function for any x in the domain of definition and any allowed values bases of a logarithm. Based on the definition of the derivative, we get:
(log a x) " = lim ∆ x → 0 log a (x + ∆ x) - log a x ∆ x = lim ∆ x → 0 log a x + ∆ x x ∆ x = = lim ∆ x → 0 1 ∆ x log a 1 + ∆ x x = lim ∆ x → 0 log a 1 + ∆ x x 1 ∆ x = = lim ∆ x → 0 log a 1 + ∆ x x 1 ∆ x x x = lim ∆ x → 0 1 x log a 1 + ∆ x x x ∆ x = = 1 x log a lim ∆ x → 0 1 + ∆ x x x ∆ x = 1 x log a e = 1 x ln e ln a = 1 x ln a
It can be seen from the specified chain of equalities that the transformations were built on the basis of the logarithm property. The equality lim ∆ x → 0 1 + ∆ x x x ∆ x = e is true in accordance with the second remarkable limit.
Example 4
Logarithmic functions are given:
f 1 (x) = log log 3 x , f 2 (x) = log x
It is necessary to calculate their derivatives.
Decision
Let's apply the derived formula:
f 1 "(x) = (log ln 3 x)" = 1 x ln (ln 3) ; f 2 "(x) \u003d (ln x)" \u003d 1 x ln e \u003d 1 x
So the derivative of the natural logarithm is one divided by x.
Derivatives of trigonometric functions
Proof 6We use some trigonometric formulas and the first remarkable limit to derive the formula for the derivative of a trigonometric function.
According to the definition of the derivative of the sine function, we get:
(sin x) " = lim ∆ x → 0 sin (x + ∆ x) - sin x ∆ x
The formula for the difference of sines will allow us to perform the following actions:
(sin x) " = lim ∆ x → 0 sin (x + ∆ x) - sin x ∆ x = = lim ∆ x → 0 2 sin x + ∆ x - x 2 cos x + ∆ x + x 2 ∆ x = = lim ∆ x → 0 sin ∆ x 2 cos x + ∆ x 2 ∆ x 2 = = cos x + 0 2 lim ∆ x → 0 sin ∆ x 2 ∆ x 2
Finally, we use the first wonderful limit:
sin "x = cos x + 0 2 lim ∆ x → 0 sin ∆ x 2 ∆ x 2 = cos x
So the derivative of the function sin x will cos x.
We will also prove the formula for the cosine derivative in the same way:
cos "x = lim ∆ x → 0 cos (x + ∆ x) - cos x ∆ x = = lim ∆ x → 0 - 2 sin x + ∆ x - x 2 sin x + ∆ x + x 2 ∆ x = = - lim ∆ x → 0 sin ∆ x 2 sin x + ∆ x 2 ∆ x 2 = = - sin x + 0 2 lim ∆ x → 0 sin ∆ x 2 ∆ x 2 = - sin x
Those. derivative cos functions x will be – sin x.
We derive the formulas for the derivatives of the tangent and cotangent based on the rules of differentiation:
t g "x = sin x cos x" = sin "x cos x - sin x cos "x cos 2 x = = cos x cos x - sin x (- sin x) cos 2 x = sin 2 x + cos 2 x cos 2 x = 1 cos 2 x c t g "x = cos x sin x" = cos "x sin x - cos x sin "x sin 2 x = = - sin x sin x - cos x cos x sin 2 x = - sin 2 x + cos 2 x sin 2 x = - 1 sin 2 x
Derivatives of inverse trigonometric functions
The section on the derivative of inverse functions gives comprehensive information on the proof of the formulas for the derivatives of the arcsine, arccosine, arctangent and arccotangent, so we will not duplicate the material here.
Derivatives of hyperbolic functions
Proof 7We can derive formulas for the derivatives of the hyperbolic sine, cosine, tangent and cotangent using the differentiation rule and the formula for the derivative of the exponential function:
s h "x = e x - e - x 2" = 1 2 e x "- e - x" == 1 2 e x - - e - x = e x + e - x 2 = c h x c h "x = e x + e - x 2" = 1 2 e x "+ e - x" == 1 2 e x + - e - x = e x - e - x 2 = s h x t h "x = s h x c h x" = s h "x c h x - s h x c h "x c h 2 x = c h 2 x - s h 2 x c h 2 x = 1 c h 2 x c t h "x = c h x s h x" = c h "x s h x - c h x s h "x s h 2 x = s h 2 x - c h 2 x s h 2 x = - 1 s h 2 x
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It is absolutely impossible to solve physical problems or examples in mathematics without knowledge about the derivative and methods for calculating it. The derivative is one of the most important concepts of 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 the derivative
Let there be a function f(x) , given in some interval (a,b) . The points x and x0 belong to this interval. When x changes, the function itself changes. Argument change - difference of its values x-x0 . This difference is written as delta x and is called argument increment. The change or increment of a function is the difference between the values of the function at two points. Derivative definition:
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 is the point in finding such a limit? But which one:
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.
The physical meaning of the derivative: the time derivative of the path is equal to the speed of the rectilinear motion.
Indeed, since school days, everyone knows that speed is a private path. x=f(t) and time t . Average speed over a certain period of time:
To find out the speed of movement at a time t0 you need to calculate the limit:
Rule one: take out the constant
The constant can be taken out of the sign of the derivative. Moreover, it must be done. When solving examples in mathematics, take as a rule - if you can simplify the expression, be sure to simplify .
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 a function:
Rule three: the 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:
Decision:
Here it is important to say about the calculation of derivatives of complex functions. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument by the derivative of the intermediate argument with respect to the independent variable.
In the above example, we encounter 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 consider 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: The derivative of the quotient of two functions
Formula for determining the derivative of a quotient of two functions:
We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it sounds, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.
With any question on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult control and deal with tasks, even if you have never dealt with the calculation of derivatives before.
It's very easy to remember.
Well, let's not go far, let's immediately consider inverse function. What 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 a “natural” one, and we use a special notation for it: we write instead.
What is equal to? Of course, .
The derivative of the natural logarithm is also very simple:
Examples:
- Find the derivative of the function.
- What is the derivative of the function?
Answers: Exhibitor and natural logarithm- functions are uniquely simple in terms 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
What rules? Another new term, again?!...
Differentiation is the process of finding the derivative.
Only and everything. What is another word for this process? Not proizvodnovanie... The differential of mathematics is called the very increment of the 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 will also need formulas for their increments:
There are 5 rules in total.
The constant is taken out of the sign of the derivative.
If - some constant number (constant), then.
Obviously, this rule also works for the difference: .
Let's prove it. Let, or easier.
Examples.
Find derivatives of functions:
- at the point;
- at the point;
- at the point;
- at the point.
Solutions:
- (the derivative is the same at all points, since it is linear function, remember?);
Derivative of a product
Everything is similar here: we introduce a new function and find its increment:
Derivative:
Examples:
- Find derivatives of functions and;
- Find the derivative of a function at a point.
Solutions:
Derivative of exponential function
Now your knowledge is enough to learn how to find the derivative of any exponential function, and not just the exponent (have you forgotten what it is yet?).
So where is some number.
We already know the derivative of the function, so let's try to bring our function to a new base:
For this we use simple rule: . Then:
Well, it worked. Now try to find the derivative, and don't forget that this function is complex.
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 factor appeared, which is just a number, but not a variable.
Examples:
Find derivatives of 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 it is left in this form.
Note that here is the quotient of two functions, so we apply the appropriate differentiation rule:
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 from the logarithm with a different base, for example, :
We need to bring this logarithm to the base. How do you change the base of a logarithm? I hope you remember this formula:
Only now instead of we will write:
The denominator turned out to be just a constant (a constant number, without a variable). The derivative is very simple:
Derivatives of exponential and logarithmic functions almost never occur in the exam, but it will not be superfluous to know them.
Derivative of a complex function.
What " complex function"? No, this is not a logarithm, and not an arc tangent. These functions can be difficult to understand (although if the logarithm seems difficult to you, read the topic "Logarithms" and everything will work out), but in terms of mathematics, the word "complex" does not mean "difficult".
Imagine a small conveyor: two people are sitting and doing some actions with some objects. For example, the first 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 opposite 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, they give us a number (chocolate), I find its cosine (wrapper), and then you square what I got (tie it with a ribbon). What happened? Function. This is an example of a complex function: when, in order to find its value, we do the first action directly with the variable, and then another second action with what happened as a 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 the order of actions changes, the function changes.
Second example: (same). .
The last action we do will be called "external" function, and the action performed 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: The separation of inner and outer functions is very similar to changing variables: for example, in the function
- What action will we take first? First we calculate the sine, and only then we raise it to a cube. So it's an internal function, not an external one.
And the original function is their composition: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: .
we change variables and get a function.
Well, now we will extract our chocolate - look for the 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. For the original example, it looks like this:
Another example:
So, let's finally formulate the official rule:
Algorithm for finding the derivative of a complex function:
It seems to be simple, right?
Let's check with examples:
Solutions:
1) Internal: ;
External: ;
2) Internal: ;
(just don’t try to reduce by now! Nothing is taken out from under the cosine, remember?)
3) Internal: ;
External: ;
It is immediately clear that there is a three-level complex function here: after all, this is already a complex function in itself, and we still extract the root from it, that is, we perform the third action (put chocolate in a wrapper and with a ribbon in a briefcase). 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 it all.
In such cases, it is convenient to number the actions. That is, let's imagine what we know. In what order will we perform actions to calculate the value of this expression? Let's look at an example:
The later the action is performed, the more "external" the corresponding function will be. The sequence of actions - as before:
Here the nesting is generally 4-level. Let's determine the course of action.
1. Radical expression. .
2. Root. .
3. Sinus. .
4. Square. .
5. Putting it all together:
DERIVATIVE. BRIEFLY ABOUT THE MAIN
Function derivative- the ratio of the increment of the function to the increment of the argument with an infinitesimal increment of the argument:
Basic derivatives:
Differentiation rules:
The constant is taken out of the sign of the derivative:
Derivative of sum:
Derivative product:
Derivative of the quotient:
Derivative of a complex function:
Algorithm for finding the derivative of a complex function:
- We define the "internal" function, find its derivative.
- We define the "external" function, find its derivative.
- We multiply the results of the first and second points.
FIRST DERIVATIVE
FIRST DERIVATIVE
(first derivative) The growth rate of the value of the function when its argument grows at some point, if the function itself is defined at this point. On the graph, the first derivative of the function shows the angle of its inclination. If a y=f(x), its first derivative at a point x0 is the limit to which f(x0+а)–f(x0)/а as a tends to an infinitesimal value. The first derivative can be denoted dy/dx or y´(x). Function y(x) has a constant value at the point x0, if dy/dx at the point x0 equals zero. The first derivative equal to zero is a necessary but not sufficient condition for the function to reach its maximum or minimum at a given point.
Economy. Dictionary. - M.: "INFRA-M", Publishing house "Ves Mir". J. Black. General editorial staff: Doctor of Economics Osadchaya I.M.. 2000 .
Economic dictionary. 2000 .
See what "FIRST DERIVATIVE" is in other dictionaries:
- (derivative) The rate at which the value of the function increments when its argument is incremented at some point, if the function itself is defined at that point. On the graph, the first derivative of the function shows the angle of its inclination. If y \u003d f (x), its first derivative at the point ... ... Economic dictionary
This term has other meanings, see Derivative. Illustration of the concept of a derivative Derivative ... Wikipedia
The derivative is the basic concept of differential calculus, which characterizes the rate of change of a function. It is defined as the limit of the ratio of the increment of a function to the increment of its argument when the increment of the argument tends to zero, if such a limit ... ... Wikipedia
Boundary problem special kind; is to find in the area D variables x=(x1,..., x n).solutions differential equation(1) of even order 2m in set values of all derivatives of order not higher than m on the boundary S of the domain D (or part of it) ... Mathematical Encyclopedia
- (second derivative) The first derivative of the first derivative of the function. The first derivative measures the slope of the function; the second derivative measures how the slope changes with increasing argument. Second derivative of y = f(x)… … Economic dictionary
This article or section needs revision. Please improve the article in accordance with the rules for writing articles. Fractional pro ... Wikipedia
- (cross partial derivative) The effect of changing one argument of a function from two or more variables on the derivative of this function, taken with respect to another argument. If y \u003d f (x, z), then its derivative, or the first derivative of the function y with respect to the argument x, is ... ... Economic dictionary
dot speed analogue- The first derivative of the point movement along the generalized coordinate of the mechanism ...
analogue of the angular velocity of the link- The first derivative of the angle of rotation of the link with respect to the generalized coordinate of the mechanism ... Polytechnic terminological explanatory dictionary
generalized speed of the mechanism- The first derivative of the generalized coordinate of the mechanism with respect to time ... Polytechnic terminological explanatory dictionary
Books
- Collection of problems in differential geometry and topology, Mishchenko A.S.
- My scientific articles Book 3. Density matrix method in quantum theories of laser, arbitrary atom, Bondarev Boris Vladimirovich. This book considers published scientific articles in which new quantum theories of the laser, an arbitrary atom, and a damped quantum oscillator are presented by the method of density matrices.…
