Derivative formula by definition. Derivative of a function

16.10.2019 -

Let the function y = f(x) be defined in the interval X. Derivative function y = f(x) at point x o is called the limit

= .

If this limit finite, then the function f(x) is called differentiable at the point x o;

Moreover, it turns out to be necessarily continuous at this point. If the limit under consideration is equal to  (or - ), then provided that the function at the point o X If the limit under consideration is equal to  (or - ), then provided that the function at the point o is continuous, we will say that the function f(x) has at the point.

infinite derivative

The derivative is denoted by the symbols

y , f (x o), , . Finding the derivative is called differentiation functions. Geometric meaning of derivative If the limit under consideration is equal to  (or - ), then provided that the function at the point o ; is that the derivative is the slope of the tangent to the curve y=f(x) at a given point physical meaning - is that the derivative of the path with respect to time is the instantaneous speed of the moving point at straight motion

s = s(t) at time t o . If With is a constant number, and u = u(x), v = v(x) are some differentiable functions, then following rules

differentiation:

1) (c) " = 0, (cu) " = cu";

2) (u+v)" = u"+v";

3) (uv)" = u"v+v"u;

4) (u/v)" = (u"v-v"u)/v 2; 5) if y = f(u), u = (x), i.e. y = f((x)) - complex function or superposition

, composed of differentiable functions  and f, then , or

6) if for a function y = f(x) there is an inverse differentiable function x = g(y), and  0, then .

Based on the definition of the derivative and the rules of differentiation, it is possible to compile a list of tabular derivatives of the main elementary functions. 1. (u )" =  u  1 u" ( ).

2. (a u)" = a u lna u".

3. (e u)" = e u u".

4. (log a u)" = u"/(u ln a).

5. (ln u)" = u"/u.

6. (sin u)" = cos u u".

7. (cos u)" = - sin u u".

8. (tg u)" = 1/ cos 2 u u".

9. (ctg u)" = - u" / sin 2 u.

10. (arcsin u)" = u" / .

11. (arccos u)" = - u" / .

12. (arctg u)" = u"/(1 + u 2).

13. (arcctg u)" = - u"/(1 + u 2). Let's calculate the derivative of the power-exponential expression y=u v , (u>0), where u And v If the limit under consideration is equal to  (or - ), then provided that the function at the point the essence of the function from , having derivatives at a given point,u".

Taking logarithms of the equality y=u v , we obtain ln y = v ln u. If the limit under consideration is equal to  (or - ), then provided that the function at the point Equating derivatives with respect to

y"/y = vu"/u +v" ln u, whence y" = y (vu"/u +v" ln u).

(u v)"=u v (vu"/u+v" ln u), u > 0.

For example, if y = x sin x, then y" = x sin x (sin x/x + cos x ln x).

If the function y = f(x) is differentiable at the point x, i.e. has a finite derivative at this point y", then = y"+, where 0 at х 0; hence  y = y" х +  x.

The main part of the function increment, linear with respect to x, is called differential functions and is denoted by dy: dy = y" х. If we put y=x in this formula, we get dx = x"х = 1х =х, therefore dy=y"dx, i.e. the symbol for The derivative notation can be thought of as a fraction.

Function increment  y is the increment of the ordinate of the curve, and the differential d y is the ordinate increment of the tangent.

Let us find for the function y=f(x) its derivative y = f (x). The derivative of this derivative is called second order derivative functions f(x), or second derivative, .

and is designated

The following are defined and designated in the same way: - ,

third order derivative

fourth order derivative - and generally speaking - .

nth order derivative.15. Example 3

Calculate the derivative of the function y=(3x 3 -2x+1)sin x. Solution.

By rule 3, y"=(3x 3 -2x+1)"sin x + (3x 3 -2x+1)(sin x)" = = (9x 2 -2)sin x + (3x 3 -2x +1)cos x. 3.16 Example

Calculate the derivative of the function y=(3x 3 -2x+1)sin x.. = .

nth order derivative.17. Find y", y = tan x + .

Calculate the derivative of the function y=(3x 3 -2x+1)sin x. Using the rules for differentiating the sum and the quotient, we obtain: y"=(tgx + )" = (tgx)" + ()" = + .

Find the derivative of the complex function y= , u=x 4 +1.

According to the rule of differentiation of a complex function, we get: y" x =y " u u" x =()" u (x 4 +1)" x =(2u +. Since u=x 4 +1, then (2 x 4 + 2+

1. In this lesson we will learn to apply formulas and rules of differentiation. Examples. Find derivatives of functions. y=x 7 +x 5 -x 4 +x 3 -x 2 +x-9. Applying the rule I, formulas

4, 2 and 1

2. . We get: 3.

y’=7x 6 +5x 4 -4x 3 +3x 2 -2x+1.

y=3x 6 -2x+5. We solve similarly, using the same formulas and formula Examples. Find derivatives of functions. y=x 7 +x 5 -x 4 +x 3 -x 2 +x-9. Applying the rule 3, 5 y’=3∙6x 5 -2=18x 5 -2. 6 y’=3∙6x 5 -2=18x 5 -2. 1.

Applying the rule And y=x 7 +x 5 -x 4 +x 3 -x 2 +x-9. Applying the rule 5 y’=3∙6x 5 -2=18x 5 -2. 1 .

Applying the rule Examples. Find derivatives of functions. IV 4 In the fifth example, according to the rule the derivative of the sum is equal to the sum of the derivatives, and we just found the derivative of the 1st term (example), therefore, we will find derivatives 2nd And 3rd terms, and

for 1st summand we can immediately write the result. y’=3∙6x 5 -2=18x 5 -2. Let's differentiate 2nd 4 3rd 4 terms according to the formula

Look at this example and the result. Did you catch the pattern? Fine. This means we have a new formula and can add it to our derivatives table.

Let's solve the sixth example and derive another formula.

Let's use the rule And and formula 4 . Let's reduce the resulting fractions.

Let's look at this function and its derivative. You, of course, understand the pattern and are ready to name the formula:

Learning new formulas!

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 .

Calculate the derivative of the function y=(3x 3 -2x+1)sin x.

New argument value x=x 0 +Δx. Let's substitute the data: 4.01=4+Δx, hence the increment of the argument Δх=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 Δх=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.

Calculate the derivative of the function y=(3x 3 -2x+1)sin x.

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=xn.

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.

Let's learn together!

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Derivative calculations are often found in Unified State Exam assignments. This page contains a list of formulas for finding derivatives.

Rules of differentiation

(k⋅ f(x))′=k⋅ f ′(x).
(f(x)+g(x))′=f′(x)+g′(x).
(f(x)⋅ g(x))′=f′(x)⋅ g(x)+f(x)⋅ g′(x).
Derivative of a complex function. If y=F(u), and u=u(x), then the function y=f(x)=F(u(x)) is called a complex function of x. Equal to y′(x)=Fu′⋅ ux′.
Derivative of an implicit function. The function y=f(x) is called an implicit function defined by the relation F(x,y)=0 if F(x,f(x))≡0.
Derivative of the inverse function. If g(f(x))=x, then the function g(x) is called inverse function for the function y=f(x).
Derivative of a parametrically defined function. Let x and y be specified as functions of the variable t: x=x(t), y=y(t). They say that y=y(x) is parametric given function on the interval x∈ (a;b), if on this interval the equation x=x(t) can be expressed as t=t(x) and the function y=y(t(x))=y(x) can be defined.
Derivative of a power-exponential function. Found by taking logarithms to the base of the natural logarithm.

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The content of the article

DERIVATIVE– derivative of the function y = f(x), given on a certain interval ( a, b) at point x of this interval is called the limit to which the ratio of the increment of the function tends f at this point to the corresponding increment of the argument when the increment of the argument tends to zero.

The derivative is usually denoted as follows:

Other designations are also widely used:

Instant speed.

Let the point M moves in a straight line. Distance s moving point, counted from some initial position M 0 , depends on time t, i.e. s there is a function of time t: s= f(t). Let at some point in time t moving point M was at a distance s from initial position M 0, and at some next moment t+D t found herself in a position M 1 - on distance s+D s from the initial position ( see pic.).

Thus, over a period of time D t distance s changed by the amount D s. In this case they say that during the time interval D t magnitude s received increment D s.

The average speed cannot in all cases accurately characterize the speed of movement of a point M at a point in time t. If, for example, the body at the beginning of the interval D t moved very quickly, and at the end very slowly, then the average speed will not be able to reflect the indicated features of the point’s movement and give an idea of the true speed of its movement at the moment t. To more accurately express the true speed using the average speed, you need to take a shorter period of time D t. Most fully characterizes the speed of movement of a point at the moment t the limit to which the average speed tends at D t® 0. This limit is called the current speed:

Thus, the speed of movement at a given moment is called the limit of the path increment ratio D s to time increment D t, when the time increment tends to zero. Because

Geometric meaning of the derivative. Tangent to the graph of a function.

The construction of tangent lines is one of those problems that led to the birth of differential calculus. The first published work related to differential calculus, written by Leibniz, was entitled New method maxima and minima, as well as tangents, for which neither fractional nor irrational quantities, and a special type of calculus for this, serve as an obstacle.

Let the curve be the graph of the function y =f(x) in a rectangular coordinate system ( cm. rice.).

At some value x function matters y =f(x). These values x And y the point on the curve corresponds M 0(x, y). If the argument x give increment D x, then the new value of the argument x+D x corresponds to the new function value y+ D y = f(x + D x). The corresponding point of the curve will be the point M 1(x+D x,y+D y). If you draw a secant M 0M 1 and denoted by j the angle formed by a transversal with the positive direction of the axis Ox, from the figure it is immediately clear that .

If now D x tends to zero, then the point M 1 moves along the curve, approaching the point M 0, and angle j changes with D x. At Dx® 0 angle j tends to a certain limit a and the straight line passing through the point M 0 and the component with the positive direction of the x-axis, angle a, will be the desired tangent. Its slope is:

Hence, f´( x) = tga

those. derivative value f´( x) at given value argument x equals the tangent of the angle formed by the tangent to the graph of the function f(x) at the corresponding point M 0(x,y) with positive axis direction Ox.

Differentiability of functions.

Definition. If the function y = f(x) has a derivative at the point x = x 0, then the function is differentiable at this point.

Continuity of a function having a derivative. Theorem.

If the function y = f(x) is differentiable at some point x = x 0, then it is continuous at this point.

Thus, the function cannot have a derivative at discontinuity points. The opposite conclusion is incorrect, i.e. from the fact that at some point x = x 0 function y = f(x) is continuous does not mean that it is differentiable at this point. For example, the function y = |x| continuous for everyone x(–Ґ x x = 0 has no derivative. At this point there is no tangent to the graph. There is a right tangent and a left one, but they do not coincide.

Some theorems on differentiable functions. Theorem on the roots of the derivative (Rolle's theorem). If the function f(x) is continuous on the segment [a,b], is differentiable at all interior points of this segment and at the ends x = a y’=3∙6x 5 -2=18x 5 -2. x = b goes to zero ( f(a) = f(b) = 0), then inside the segment [ a,b] there is at least one point x= If, a c b, in which the derivative fў( x) goes to zero, i.e. fў( c) = 0.

Finite increment theorem (Lagrange's theorem). If the function f(x) is continuous on the interval [ a, b] and is differentiable at all interior points of this segment, then inside the segment [ a, b] there is at least one point If, a c b that

f(b) – f(a) = fў( c)(b– a).

Theorem on the ratio of the increments of two functions (Cauchy's theorem). If f(x) And g(x) – two functions continuous on the segment [a, b] and differentiable at all interior points of this segment, and gў( x) does not vanish anywhere inside this segment, then inside the segment [ a, b] there is such a point x = If, a c b that

Derivatives of various orders.

Let the function y =f(x) is differentiable on some interval [ a, b]. Derivative values f ў( x), generally speaking, depend on x, i.e. derivative f ў( x) is also a function of x. When differentiating this function, we obtain the so-called second derivative of the function f(x), which is denoted f ўў ( x).

Derivative n- th order of function f(x) is called the derivative (first order) of the derivative n- 1- th and is denoted by the symbol y(n) = (y(n– 1))ў.

Differentials of various orders.

Function differential y = f(x), Where x– independent variable, yes dy = f ў( x)dx, some function from x, but from x only the first factor can depend f ў( x), the second factor ( dx) is the increment of the independent variable x and does not depend on the value of this variable. Because dy there is a function from x, then we can determine the differential of this function. The differential of the differential of a function is called the second differential or second-order differential of this function and is denoted d 2y:

d(dx) = d 2y = f ўў( x)(dx) 2 .

Differential n- of the first order is called the first differential of the differential n- 1- th order:

d n y = d(d n–1y) = f(n)(x)dx(n).

Partial derivative.

If a function depends not on one, but on several arguments x i(i varies from 1 to n,i= 1, 2,… n),f(x 1,x 2,… x n), then in differential calculus the concept of partial derivative is introduced, which characterizes the rate of change of a function of several variables when only one argument changes, for example, x i. 1st order partial derivative with respect to x i is defined as an ordinary derivative, and it is assumed that all arguments except x i, save constant values. For partial derivatives, the notation is introduced

The 1st order partial derivatives defined in this way (as functions of the same arguments) can, in turn, also have partial derivatives, these are second order partial derivatives, etc. Such derivatives taken from different arguments are called mixed. Continuous mixed derivatives of the same order do not depend on the order of differentiation and are equal to each other.

Anna Chugainova

Create a ratio and calculate the limit.

Where did it come from? table of derivatives and differentiation rules? Thanks to the only limit. It seems like magic, but in reality it is sleight of hand and no fraud. At the lesson What is a derivative? I started looking at specific examples, where, using the definition, I found the derivatives of linear and quadratic function. For the purpose of cognitive warm-up, we will continue to disturb table of derivatives, honing the algorithm and technical solutions:

Example 1

Essentially, you need to prove a special case of the derivative of a power function, which usually appears in the table: .

Solution technically formalized in two ways. Let's start with the first, already familiar approach: the ladder starts with a plank, and the derivative function starts with the derivative at a point.

Let's consider some(specific) point belonging to domain of definition function in which there is a derivative. Let us set the increment at this point (of course, not going beyondo/o -I) and compose the corresponding increment of the function:

Let's calculate the limit:

The uncertainty 0:0 is eliminated by a standard technique, considered back in the first century BC. Multiply the numerator and denominator by the conjugate expression :

The technique for solving such a limit is discussed in detail in the introductory lesson. about the limits of functions.

Since you can choose ANY point of the interval as quality, then, by making the replacement, we get:

Answer

Once again let's rejoice at logarithms:

Example 2

Find the derivative of a function using the definition of derivative

Solution: Let's consider a different approach to promoting the same task. It is exactly the same, but more rational in terms of design. The idea is to get rid of the subscript at the beginning of the solution and use the letter instead of the letter.

Let's consider arbitrary point belonging to domain of definition function (interval) and set the increment in it. But here, by the way, as in most cases, you can do without any reservations, since the logarithmic function is differentiable at any point in the domain of definition.

Then the corresponding increment of the function is:

Let's find the derivative:

The simplicity of the design is balanced by the confusion that may arise for beginners (and not only). After all, we are used to the fact that the letter “X” changes in the limit! But here everything is different: - an antique statue, and - a living visitor, briskly walking along the corridor of the museum. That is, “x” is “like a constant.”

I will comment on the elimination of uncertainty step by step:

(1) We use the property of the logarithm.

(2) In parentheses, divide the numerator by the denominator term by term.

(3) In the denominator, we artificially multiply and divide by “x” to take advantage of remarkable limit , while as infinitesimal stands out.

Answer: by definition of derivative:

Or in short:

I propose to construct two more table formulas yourself:

Example 3

In this case, it is convenient to immediately reduce the compiled increment to a common denominator. An approximate sample of the assignment at the end of the lesson (first method).

Example 3:Solution : consider some point , belonging to the domain of definition of the function . Let us set the increment at this point and compose the corresponding increment of the function:

Let's find the derivative at the point :

Since as a you can select any point function domain , That And
Answer : by definition of derivative

Example 4

Find derivative by definition

And here everything needs to be reduced to wonderful limit. The solution is formalized in the second way.

A number of other tabular derivatives. Full list can be found in a school textbook, or, for example, the 1st volume of Fichtenholtz. I don’t see much point in copying proofs of differentiation rules from books - they are also generated by the formula.

Example 4:Solution , belonging to , and set the increment in it

Let's find the derivative:

Using a wonderful limit

Answer : a-priory

Example 5

Find the derivative of a function using the definition of derivative

Solution: we use the first design style. Let's consider some point belonging to , and specify the increment of the argument at it. Then the corresponding increment of the function is:

Perhaps some readers have not yet fully understood the principle by which increments need to be made. Take a point (number) and find the value of the function in it: , that is, into the function instead of"X" should be substituted. Now we also take a very specific number and also substitute it into the function instead of"iksa": . We write down the difference, and it is necessary put in brackets completely.

Compiled function increment It can be beneficial to immediately simplify. For what? Facilitate and shorten the solution to a further limit.

We use formulas, open the brackets and shorten everything that can be shortened:

The turkey is gutted, no problem with the roast:

Since we can choose any real number as a value, we make the replacement and get .

Answer: a-priory.

For verification purposes, let’s find the derivative using differentiation rules and tables:

It is always useful and pleasant to know the correct answer in advance, so it is better to differentiate the proposed function in a “quick” way, either mentally or in a draft, at the very beginning of the solution.

Example 6

Find the derivative of a function by definition of derivative

This is an example for independent decision. The result is obvious:

Example 6:Solution : consider some point , belonging to , and set the increment of the argument in it . Then the corresponding increment of the function is:

Let's calculate the derivative:

Thus:
Because as you can choose any real number, then And
Answer : a-priory.

Let's go back to style #2:

Example 7

Let's find out immediately what should happen. By rule of differentiation of complex functions:

Solution: consider an arbitrary point belonging to , set the increment of the argument at it and compose the increment of the function:

Let's find the derivative:

(1) Use trigonometric formula .

(2) Under the sine we open the brackets, under the cosine we present similar terms.

(3) Under the sine we reduce the terms, under the cosine we divide the numerator by the denominator term by term.

(4) Due to the oddness of the sine, we take out the “minus”. Under the cosine we indicate that the term .

(5) We carry out artificial multiplication in the denominator in order to use first wonderful limit. Thus, the uncertainty is eliminated, let’s tidy up the result.

Answer: a-priory

As you can see, the main difficulty of the problem under consideration rests on the complexity of the limit itself + a slight uniqueness of the packaging. In practice, both methods of design occur, so I describe both approaches in as much detail as possible. They are equivalent, but still, in my subjective impression, it is more advisable for dummies to stick to option 1 with “X-zero”.

Example 8

Using the definition, find the derivative of the function

Example 8:Solution : consider an arbitrary point , belonging to , let us set the increment in it and compose the increment of the function:

Let's find the derivative:

We use the trigonometric formula and the first remarkable limit:

Answer : a-priory

Let's look at a rarer version of the problem:

Example 9

Find the derivative of the function at the point using the definition of derivative.

Firstly, what should be the bottom line? Number

Let's calculate the answer in the standard way:

Solution: from a clarity point of view, this task is much simpler, since the formula instead considers a specific value.

Let's set the increment at the point and compose the corresponding increment of the function:

Let's calculate the derivative at a point:

We use a very rare tangent difference formula and once again we reduce the solution to the first wonderful limit:

Answer: by definition of derivative at a point.

The problem is not so difficult to solve and “in general view» – it is enough to replace with or simply depending on the design method. In this case, it is clear that the result will not be a number, but a derived function.

Example 10

Using the definition, find the derivative of the function at a point (one of which may turn out to be infinite), which I’m talking about general outline already told on theoretical lesson about derivative.

Some piecewise given functions are also differentiable at the “junction” points of the graph, for example, catdog has a common derivative and a common tangent (x-axis) at the point. Curve, but differentiable by ! Those interested can verify this for themselves using the example just solved.