Operations with derivatives. What is a derivative? Definition and meaning of a derivative function. Common notations for the derivative of a function at a point

Concept of derivative

Let the function f(x) is defined on some interval X. Let's give the value of the argument at the point x 0 X arbitrary increment Δ x so that the point x 0 + Δ x also belonged to X. Then the corresponding increment of function f(x) will be Δ at = f(x 0 + Δ x) - f(x 0).

Definition 1. Derivative of function f(x) at the point x 0 is called the limit of the ratio of the increment of the function at this point to the increment of the argument at Δ x 0 (if this limit exists).

To denote the derivative of a function, we use the symbols y" (x 0) or f"(x 0):

If at some point x 0 limit (4.1) is infinite:

then they say that at the point x 0 function f(x) has infinite derivative.

If the function f(x) has a derivative at each point of the set X, then the derivative f"(x) is also a function of the argument X, defined on X.

Geometric meaning of derivative

To clarify the geometric meaning of the derivative, we need to determine the tangent to the graph of the function at a given point.

Definition 2. Tangent to the graph of the function y = f(x) at point M called the limit position of the secant MN, when is the point N tends to a point M along the curve f(x).

Let the point M on the curve f(x) corresponds to the value of the argument x 0, and point N- argument value x 0 + Δ x(Fig. 4.1). From the definition of a tangent it follows that for its existence at a point x 0 it is necessary that there is a limit that equal to angle inclination of the tangent to the axis Oh. From a triangle M.N.A. it follows that

If the derivative of the function f(x) at point x 0 exists, then according to (4.1), we obtain

From this follows a clear conclusion that derivative f"(x 0) equal to the angular coefficient (tangent of the angle of inclination to the positive direction of the Ox axis) of the tangent to the graph of the function y = f(x) V point M(x 0, f(x 0)). In this case, the angle of the tangent is determined from formula (4.2):

Physical meaning of the derivative

Let's assume that the function l = f(t) describes the law of motion material point in a straight line as path dependence l from time to time t. Then the difference Δ l = f(t +Δ t) - f(t) - is the path traveled during the time interval Δ t, and the ratio Δ l/Δ t- average speed over time Δ t. Then the limit determines instantaneous point speed at a point in time t as the derivative of the path with respect to time.

In a certain sense, the derivative of the function at = f(x) can also be interpreted as the rate of change of a function: the greater the value f"(x), the greater the angle of inclination of the tangent to the curve, the steeper the graph f(x) and the function grows faster.

Right and left derivatives

By analogy with the concepts of one-sided limits of a function, the concepts of right and left derivatives of a function at a point are introduced.

Definition 3. Right (left) derivative of a function at = f(x) at the point x 0 is called the right (left) limit of relation (4.1) for Δ x 0 if this limit exists.

The following symbolism is used to denote one-sided derivatives:

If the function f(x) has at the point x 0 derivative, then it has left and right derivatives at this point, which coincide.

Let's give an example of a function that has one-sided derivatives at a point that are not equal to each other. This f(x) = |x|. Indeed, at the point x = 0 we have f' +(0) = 1, f" -(0) = -1 (Fig. 4.2) and f' +(0) ≠ f’ -(0), i.e. the function has no derivative at X = 0.

The operation of finding the derivative of a function is called it differentiation; a function that has a derivative at a point is called differentiable.

The connection between differentiability and continuity of a function at a point is established by the following theorem.

THEOREM 1 . If a function is differentiable at a point x 0, then it is continuous at this point.

The converse is not true: function f(x), continuous at a point, may not have a derivative at that point. Such an example is the function at = |x|; it is continuous at a point x= 0, but has no derivative at this point.

Thus, the requirement of differentiability of a function is stronger than the requirement of continuity, since the second automatically follows from the first.

Equation of the tangent to the graph of a function at a given point

As stated in Section 3.9, the equation of a line passing through a point M(x 0, y 0) with slope k looks like

Let the function be given at = f(x). Then since its derivative at some point M(x 0, y 0) is the slope of the tangent to the graph of this function at the point M, then it follows that the equation of the tangent to the graph of the function f(x) at this point has the form

The derivative of a function $y = f(x)$ at a given point $x_0$ is the limit of the ratio of the increment of a function to the corresponding increment of its argument, provided that the latter tends to zero:

$f"(x_0)=(lim)↙(△x→0)(△f(x_0))/(△x)$

Differentiation is the operation of finding the derivative.

Table of derivatives of some elementary functions

Basic rules of differentiation

1. The derivative of the sum (difference) is equal to the sum (difference) of the derivatives

$(f(x) ± g(x))"= f"(x)±g"(x)$

Find the derivative of the function $f(x)=3x^5-cosx+(1)/(x)$

The derivative of a sum (difference) is equal to the sum (difference) of derivatives.

$f"(x) = (3x^5)"-(cos x)" + ((1)/(x))" = 15x^4 + sinx - (1)/(x^2)$

2. Derivative of the product

$(f(x) g(x)"= f"(x) g(x)+ f(x) g(x)"$

Find the derivative $f(x)=4x cosx$

$f"(x)=(4x)"·cosx+4x·(cosx)"=4·cosx-4x·sinx$

3. Derivative of the quotient

$((f(x))/(g(x)))"=(f"(x) g(x)-f(x) g(x)")/(g^2(x)) $

Find the derivative $f(x)=(5x^5)/(e^x)$

$f"(x)=((5x^5)"·e^x-5x^5·(e^x)")/((e^x)^2)=(25x^4·e^x- 5x^5 e^x)/((e^x)^2)$

4. The derivative of a complex function is equal to the product of the derivative of the external function and the derivative of the internal function

$f(g(x))"=f"(g(x)) g"(x)$

$f"(x)=cos"(5x)·(5x)"=-sin(5x)·5= -5sin(5x)$

Physical meaning of the derivative

If a material point moves rectilinearly and its coordinate changes depending on time according to the law $x(t)$, then the instantaneous speed of this point is equal to the derivative of the function.

The point moves along the coordinate line according to the law $x(t)= 1.5t^2-3t + 7$, where $x(t)$ is the coordinate at time $t$. At what point in time will the speed of the point be equal to $12$?

1. Speed ​​is the derivative of $x(t)$, so let’s find the derivative of the given function

$v(t) = x"(t) = 1.5 2t -3 = 3t -3$

2. To find at what point in time $t$ the speed was equal to $12$, we create and solve the equation:

Geometric meaning of derivative

Recall that the equation of a line that is not parallel to the coordinate axes can be written in the form $y = kx + b$, where $k$ is the slope of the line. The coefficient $k$ is equal to the tangent of the angle of inclination between the straight line and the positive direction of the $Ox$ axis.

The derivative of the function $f(x)$ at the point $х_0$ is equal to the slope $k$ of the tangent to the graph at this point:

Therefore, we can create a general equality:

$f"(x_0) = k = tanα$

In the figure, the tangent to the function $f(x)$ increases, therefore the coefficient $k > 0$. Since $k > 0$, then $f"(x_0) = tanα > 0$. The angle $α$ between the tangent and the positive direction $Ox$ is acute.

In the figure, the tangent to the function $f(x)$ decreases, therefore, the coefficient $k< 0$, следовательно, $f"(x_0) = tgα < 0$. Угол $α$ между касательной и положительным направлением оси $Ох$ тупой.

In the figure, the tangent to the function $f(x)$ is parallel to the $Ox$ axis, therefore, the coefficient $k = 0$, therefore, $f"(x_0) = tan α = 0$. The point $x_0$ at which $f "(x_0) = 0$, called extremum.

The figure shows a graph of the function $y=f(x)$ and a tangent to this graph drawn at the point with the abscissa $x_0$. Find the value of the derivative of the function $f(x)$ at point $x_0$.

The tangent to the graph increases, therefore, $f"(x_0) = tan α > 0$

In order to find $f"(x_0)$, we find the tangent of the angle of inclination between the tangent and the positive direction of the $Ox$ axis. To do this, we build the tangent to the triangle $ABC$.

Let's find the tangent of the angle $BAC$. (The tangent of an acute angle in a right triangle is the ratio of the opposite side to the adjacent side.)

$tg BAC = (BC)/(AC) = (3)/(12)= (1)/(4)=$0.25

$f"(x_0) = tg BAC = 0.25$

Answer: $0.25$

The derivative is also used to find the intervals of increasing and decreasing functions:

If $f"(x) > 0$ on an interval, then the function $f(x)$ is increasing on this interval.

If $f"(x)< 0$ на промежутке, то функция $f(x)$ убывает на этом промежутке.

The figure shows the graph of the function $y = f(x)$. Find among the points $х_1,х_2,х_3...х_7$ those points at which the derivative of the function is negative.

In response, write down the number of these points.

Plan:

1. Derivative of a function

2. Differential function

3. Application of differential calculus to the study of functions

Derivative of a function of one variable

Let the function be defined on a certain interval. We give the argument an increment: , then the function will receive an increment. Let's find the limit of this ratio at If this limit exists, then it is called the derivative of the function. The derivative of a function has several notations: . Sometimes in the notation of a derivative, an index is used, indicating which variable the derivative is taken with respect to.

Definition. The derivative of a function at a point is the limit of the ratio of the increment of the function to the increment of the argument when the increment of the argument tends to zero (if this limit exists):

Definition. A function that has a derivative at each point of the interval is called differentiable in this interval.

Definition. The operation of finding the derivative of a function is called differentiation.

The value of the derivative of a function at a point is indicated by one of the symbols: .

Example. Find the derivative of a function at an arbitrary point.

Solution. We give the value an increment. Let's find the increment of the function at the point: . Let's create a relationship. Let's move to the limit: . Thus, .

Mechanical meaning of derivative. Since or, i.e. speed rectilinear motion of a material point at an instant of time is the derivative of the path with respect to time. This is mechanical meaning of derivative .

If a function describes any physical process, then the derivative is the rate of this process. This is physical meaning of derivative .

Geometric meaning of derivative. Consider a graph of a continuous curve that has a non-vertical tangent at a point. Let's find its angular coefficient, where is the tangent angle with the axis. To do this, draw a secant line through the point and graph (Figure 1).

Let us denote by - the angle between the secant and the axis. The figure shows that the angular coefficient of the secant is equal to

When, due to the continuity of the function, the increment also tends to zero; therefore, the point indefinitely approaches the point along the curve, and the secant, turning around the point, becomes a tangent. Angle, i.e. . Therefore, , therefore the slope of the tangent is equal to .

Slope of a tangent to a curve

We rewrite this equality in the form: , i.e. the derivative at a point is equal to the slope of the tangent to the graph of the function at the point whose abscissa is equal to . This is geometric meaning derivative .

If the point of tangency has coordinates (Figure 2), the angular coefficient of the tangent is equal to: .

Equation of a line passing through given point in a given direction has the form: .

Then tangent equation is written in the form: .

Definition. A straight line perpendicular to the tangent at the point of contact is called normal to the curve.

The angular coefficient of the normal is equal to: (since the normal is perpendicular to the tangent).

The normal equation has the form:, If .

Substituting the found values, we obtain the tangent equations, i.e. .

Normal equation: or .

If a function has a finite derivative at a point, then it is differentiable at that point. If a function is differentiable at every point of an interval, then it is differentiable in that interval.

Theorem 6.1 If a function is differentiable at some point, then it is continuous there.

The converse theorem is not true. Continuous function may not have a derivative.

Example. The function is continuous over the interval (Figure 3).

Solution.

The derivative of this function is equal to:

At a point - the function is not differentiable.

Comment. In practice, most often you have to find derivatives of complex functions. Therefore, in the table of differentiation formulas, the argument is replaced by an intermediate argument.

Derivatives table

Constant

Power function:

2) in particular;

Exponential function:

3) in particular;

Logarithmic function:

4) in particular;

Trigonometric functions:

Reverse trigonometric functions , , , :

To differentiate a function means to find its derivative, that is, to calculate the limit: . However, determining the limit in most cases is a cumbersome task.

If you know the derivatives of basic elementary functions and know the rules for differentiating results arithmetic operations over these functions, then you can easily find the derivatives of any elementary functions, according to the rules for determining derivatives, well known from the school course.

Let the functions and be two functions differentiable in a certain interval.

Theorem 6.2 The derivative of the sum (difference) of two functions is equal to the sum (difference) of the derivatives of these functions: .

The theorem is valid for any finite number of terms.

Example. Find the derivative of the function.

Solution.

Theorem 6.3 The derivative of the product of two functions is equal to the product of the derivative of the first factor and the second plus the product of the first factor and the derivative of the second: .

Example. Find the derivative of a function .

Solution.

Theorem 6.4 The derivative of the quotient of two functions, if equal to a fraction, the numerator of which is the difference between the products of the denominator of the fraction and the derivative of the numerator and the numerator of the fraction and the derivative of the denominator, and the denominator is the square of the former denominator: .

Example. Find the derivative of a function .

Solution. .

To find the derivative of a complex function, you need to multiply the derivative of this function with respect to the intermediate argument by the derivative of the intermediate argument with respect to the independent argument

This rule remains in effect if there are several intermediate arguments. So, if , , , then

Let and, then - complex function with an intermediate argument and an independent argument.

Theorem 6.5 If a function has a derivative at a point, and a function has a derivative at the corresponding point, then a complex function has a derivative at a point, which is found by the formula. , Find the derivative of the function given by the equation: .

Solution. The function is specified implicitly. Let's differentiate the equation with respect to , remembering that: . Then we find: .

Let the function be defined at a point and some of its neighborhood. Let's give the argument an increment such that the point falls into the domain of definition of the function. The function will then be incremented.

DEFINITION. Derivative of a function at a point is called the limit of the ratio of the increment of the function at this point to the increment of the argument, at (if this limit exists and is finite), i.e.

Denote: ,,,.

Derivative of a function at a point on the right (left) called

(if this limit exists and is finite).

Designated by: , – derivative at the point on the right,

, is the derivative at the point on the left.

Obviously, the following theorem is true.

THEOREM. A function has a derivative at a point if and only if at this point the derivatives of the function on the right and left exist and are equal to each other. Moreover

The following theorem establishes a connection between the existence of a derivative of a function at a point and the continuity of the function at that point.

THEOREM (a necessary condition for the existence of a derivative of a function at a point). If a function has a derivative at a point, then the function at that point is continuous.

PROOF

Let it exist. Then

,

where is infinitesimal at.

Comment

derivative of a function and denote

differentiation of function .

GEOMETRICAL AND PHYSICAL MEANING

1) Physical meaning of the derivative. If a function and its argument are physical quantities, then the derivative is the rate of change of a variable relative to the variable at a point. For example, if is the distance traveled by a point in time, then its derivative is the speed at the moment of time. If is the amount of electricity flowing through the cross section of the conductor at an instant of time, then is the rate of change in the amount of electricity at an instant of time, i.e. current strength at a moment in time.

2) Geometric meaning of derivative.

Let be some curve, be a point on the curve.

Any straight line intersecting at least two points is called secant .

Tangent to a curve at a point the limit position of a secant is called if the point tends to, moving along a curve.

From the definition it is obvious that if a tangent to a curve at a point exists, then it is the only one

Consider a curve (i.e. a graph of a function). Let it have a non-vertical tangent at a point. Its equation: (equation of a straight line passing through a point and having an angular coefficient).

By definition of the slope

where is the angle of inclination of the straight line to the axis.

Let be the angle of inclination of the secant to the axis, where. Since is a tangent, then when

Hence,

Thus, we got that – angular coefficient of the tangent to the graph of the function at the point(geometric meaning of the derivative of a function at a point). Therefore, the equation of the tangent to the curve at a point can be written in the form

Comment . A straight line passing through a point perpendicular to the tangent drawn to the curve at the point is called normal to the curve at the point . Since the angular coefficients of perpendicular straight lines are related by the relation, the equation of the normal to the curve at a point will have the form

, If .

If , then the tangent to the curve at the point will have the form

and normal.

TANGENT AND NORMAL EQUATIONS

Tangent equation

Let the function be given by the equation y=f(x), you need to write the equation tangent at the point x 0. From the definition of derivative:

y/(x)=limΔ x→0Δ yΔ x

Δ y=f(x+Δ x)−f(x).

Equation tangent to the function graph: y=kx+b (k,b=const). From the geometric meaning of the derivative: f/(x 0)=tgα= k Because x 0 and f(x 0)∈ straight line, then the equation tangent is written as: y−f(x 0)=f/(x 0)(x−x 0) , or

y=f/(x 0)· x+f(x 0)−f/(x 0)· x 0.

Normal equation

Normal- is perpendicular to tangent(see picture). Based on this:

tgβ= tg(2π−α)= ctgα=1 tgα=1 f/(x 0)

Because the angle of inclination of the normal is angle β1, then we have:

tgβ1= tg(π−β)=− tgβ=−1 f/(x).

Point ( x 0,f(x 0))∈ normal, the equation takes the form:

y−f(x 0)=−1f/(x 0)(x−x 0).

PROOF

Let it exist. Then

,

where is infinitesimal at.

But this means that it is continuous at a point (see the geometric definition of continuity). ∎

Comment . The continuity of a function at a point is not a sufficient condition for the existence of a derivative of this function at a point. For example, a function is continuous, but has no derivative at a point. Really,

and therefore does not exist.

Obviously, correspondence is a function defined on a certain set. They call her derivative of a function and denote

The operation of finding for a function its derivative function is called differentiation of function .

Derivative of sum and difference

Let functions f(x) and g(x) be given whose derivatives are known to us. For example, you can take elementary functions which are discussed above. Then you can find the derivative of the sum and difference of these functions:

(f + g)’ = f ’ + g ’

(f − g)’ = f ’ − g ’

So, the derivative of the sum (difference) of two functions is equal to the sum (difference) of the derivatives. There may be more terms. For example, (f + g + h)’ = f’ + g’ + h’.

Strictly speaking, there is no concept of “subtraction” in algebra. There is a concept of “negative element”. Therefore, the difference f − g can be rewritten as the sum f + (−1) g, and then only one formula remains - the derivative of the sum.

IN coordinate plane xOy consider the graph of the function y=f(x). Let's fix the point M(x 0 ; f (x 0)). Let's add an abscissa x 0 increment Δx. We will get a new abscissa x 0 +Δx. This is the abscissa of the point N, and the ordinate will be equal f (x 0 +Δx). The change in the abscissa entailed a change in the ordinate. This change is called the function increment and is denoted Δy.

Δy=f (x 0 +Δx) - f (x 0). Through dots M And N let's draw a secant MN, which forms an angle φ with positive axis direction Oh. Let's determine the tangent of the angle φ from right triangle MPN.

Let Δx tends to zero. Then the secant MN will tend to take a tangent position MT, and the angle φ will become an angle α . So, the tangent of the angle α is the limiting value of the tangent of the angle φ :

The limit of the ratio of the increment of a function to the increment of the argument, when the latter tends to zero, is called the derivative of the function at a given point:

Geometric meaning of derivative lies in the fact that the numerical derivative of the function at a given point is equal to the tangent of the angle formed by the tangent drawn through this point to the given curve and the positive direction of the axis Oh:

Examples.

1. Find the increment of the argument and the increment of the function y= x 2, if the initial value of the argument was equal to 4 , and new - 4,01 .

Solution.

New argument value x=x 0 +Δx. Let's substitute the data: 4.01=4+Δx, hence the increment of the argument Δx=4.01-4=0.01. The increment of a function, by definition, is equal to the difference between the new and previous values ​​of the function, i.e. Δy=f (x 0 +Δx) - f (x 0). Since we have a function y=x2, That Δу=(x 0 +Δx) 2 - (x 0) 2 =(x 0) 2 +2x 0 · Δx+(Δx) 2 - (x 0) 2 =2x 0 · Δx+(Δx) 2 =

2 · 4 · 0,01+(0,01) 2 =0,08+0,0001=0,0801.

Answer: argument increment Δx=0.01; function increment Δу=0,0801.

The function increment could be found differently: Δy=y (x 0 +Δx) -y (x 0)=y(4.01) -y(4)=4.01 2 -4 2 =16.0801-16=0.0801.

2. Find the angle of inclination of the tangent to the graph of the function y=f(x) at the point x 0, If f "(x 0) = 1.

Solution.

The value of the derivative at the point of tangency x 0 and is the value of the tangent of the tangent angle (the geometric meaning of the derivative). We have: f "(x 0) = tanα = 1 → α = 45°, because tg45°=1.

Answer: the tangent to the graph of this function forms an angle with the positive direction of the Ox axis equal to 45°.

3. Derive the formula for the derivative of the function y=x n.

Differentiation is the action of finding the derivative of a function.

When finding derivatives, use formulas that were derived based on the definition of a derivative, in the same way as we derived the formula for the derivative degree: (x n)" = nx n-1.

These are the formulas.

Table of derivatives It will be easier to memorize by pronouncing verbal formulations:

1. The derivative of a constant quantity is zero.

2. X prime is equal to one.

3. The constant factor can be taken out of the sign of the derivative.

4. The derivative of a degree is equal to the product of the exponent of this degree by a degree with the same base, but the exponent is one less.

5. The derivative of a root is equal to one divided by two equal roots.

6. The derivative of one divided by x is equal to minus one divided by x squared.

7. The derivative of the sine is equal to the cosine.

8. The derivative of the cosine is equal to minus sine.

9. The derivative of the tangent is equal to one divided by the square of the cosine.

10. The derivative of the cotangent is equal to minus one divided by the square of the sine.

We teach differentiation rules.

1. The derivative of an algebraic sum is equal to the algebraic sum of the derivatives of the terms.

2. The derivative of a product is equal to the product of the derivative of the first factor and the second plus the product of the first factor and the derivative of the second.

3. The derivative of “y” divided by “ve” is equal to a fraction in which the numerator is “y prime multiplied by “ve” minus “y multiplied by ve prime”, and the denominator is “ve squared”.

4. A special case of the formula 3.