Differential x. Differential function. The geometric meaning of the differential of a function

The differential of an argument is its increment dx = ∆ x .

The differential of a function is the product of the derivative and the increment of the argument dy = f ′( x )∙∆ x or dy = f ′( x )∙ dx .

Comment:

Comparison of an incremental differential.

Let be ∆ y and ∆x are of the same order of smallness.

Dy and ∆x are of the same order of smallness, i.e., dy and ∆y are of the same order of smallness.

α ∙ ∆x is infinitesimal of a higher order of smallness than ∆x.

.The differential is the main part of the increment of the function .

The differential of a function differs from the increment of a function by an infinitesimal

of a higher order than the increment of the argument.

The geometric meaning of the differential of a function.

dy = f ′ (x) ∙ ∆x = tgφ ∙ ∆x = NT.

The differential is equal to the increment of the ordinate of the tangent.

Differential properties.

The differential of the sum is equal to the sum of the differentials.

d ( u + v) = du + dv.

Differential product d ( u v ) = du ∙ v + u dv .

Differential of a complex function.

y = f (u), u = φ (x), dy = y ′ x dx =

dy = f ′( u ) du - invariance of the form of the differential.

Higher-order differentials.

dy = f ′(x)∙ dx, from here

Hyperbolic functions.

In many applications of mathematical analysis, there are combinations of exponential functions.

Definitions.

From the definitions of hyperbolic functions, the following relations follow:

ch 2 x – sh 2 x = 1, sh2x = 2shx ∙ chx, ch2x = ch 2 x + sh 2 x, sh (α ± β) = shαchβ ± chαshβ. Derivatives of hyperbolic functions.

Rolle's theorem.

If the function f ( x ) is defined and continuous on the closed interval [ a , b ], has a derivative at all internal points of this interval and takes equal values ​​at the ends of the interval, then inside the interval there is at least one such pointx = ξ such that f ′(ξ) = 0.

Geometric meaning.

y

f(a) = f(b), k cas = 0.

ACBOn a smooth arc [a, b] there is such a point

f(a) f(b) C, where the tangent is parallel to the chord.

a ξ b x

Lagrange's theorem (1736-1813, France).

If the function is defined and continuous on the closed interval [ a , b ] and has a derivative at all interior points of this interval, then inside this interval there is at least one point x = ξ such thatf ( b ) – f ( a ) = f ′(ξ)∙( b – a ).

The geometric meaning of Lagrange's theorem.

AND We draw a smooth arc AB.

On a smooth arc AB, there is a point C at which the tangent line is parallel to the chord AB.

Proof. Consider the function F(x) = f(x) – λ x. We choose λ so that the conditions of Rolle's theorem are satisfied.

F (x) - is defined and continuous on [ a, b], since the function f(x),.

F′(x) = f ′(x) – λ - exists,

We choose λ so that the conditions F(a) = F(b), those. f(a) – λ a = f(b) – λ b,

By Rolle's theorem, there is such a point x = ξЄ( a, b), what F′(ξ) = 0, i.e.

Increasing and decreasing functions.

The function is called increasing, if the larger value of the argument corresponds to the larger value of the function.

If the function differentiable at the point , then its increment can be represented as the sum of two terms

... These terms are infinitesimal functions for
. The first term is linear with respect to
, the second is infinitely small of a higher order than
.Really,

.

Thus, the second term for
tends to zero faster and when finding the increment of the function
the first term plays the main role
or (since
)
.

Definition . The main part of the increment function
at the point , linear with respect to
,called differential functions at this point and is denoteddyordf(x)

. (2)

Thus, we can conclude: the differential of the independent variable coincides with its increment, that is
.

Relation (2) now takes the form

(3)

Comment ... For brevity, formula (3) is often written as

(4)

The geometric meaning of the differential

Consider the graph of the differentiable function
... Points
and belong to the function graphics. At the point M drawn tangent TO to the graph of the function, the angle of which with the positive direction of the axis
denote by
... Let's draw straight MN parallel to the axis Ox and
parallel to the axis Oy... The increment of the function is equal to the length of the segment
... From a right triangle
, in which
, we get

The above reasoning allows us to conclude:

Differential function
at the point is depicted by incrementing the ordinate of the tangent to the graph of this function at its corresponding point
.

Differential-derivative connection

Consider the formula (4)

.

We divide both sides of this equality by dx, then

.

Thus, the derivative of a function is equal to the ratio of its differential to the differential of the independent variable.

Often this attitude is regarded simply as a symbol denoting the derivative of a function at by argument NS.

Convenient derivative notation is also:

,
etc.

Records are also used

,
,

especially convenient when a derivative of a complex expression is taken.

2. Differential of sum, product and quotient.

Since the differential is obtained from the derivative by multiplying it by the differential of the independent variable, then, knowing the derivatives of basic elementary functions, as well as the rules for finding derivatives, one can come to similar rules for finding differentials.

1 0 . Differential constant is zero

.

2 0 . The differential of the algebraic sum of a finite number of differentiable functions is equal to the algebraic sum of the differentials of these functions

3 0 . The differential of the product of two differentiable functions is equal to the sum of the products of the first function by the differential of the second and the second function by the differential of the first

.

Consequence. A constant multiplier can be taken out of the differential sign

.

Example... Find the differential of the function.

Solution: Let us write this function as

,

then we get

.

4. Functions given parametrically, their differentiation.

Definition . Function
is called parametrically given if both variables NS and at each are defined separately as single-valued functions of the same auxiliary variable - the parametert:

wheretvaries within
.

Comment ... Parametric setting of functions is widely used in theoretical mechanics, where the parameter t denotes time, and the equations
represent the laws of change in the projections of a moving point
on the axis
and
.

Comment ... Let us give the parametric equations of a circle and an ellipse.

a) A circle with a center at the origin and a radius r has parametric equations:

where
.

b) Let us write the parametric equations for the ellipse:

where
.

By excluding the parameter t from the parametric equations of the lines under consideration, one can come to their canonical equations.

Theorem ... If the function y from argument x is given parametrically by the equations
, where
and
differentiable bytfunctions and
, then

.

Example... Find the derivative of a function at from NS given by parametric equations.

Solution.
.

Application

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=

Definition of the differential

Consider the function \ (y = f \ left (x \ right), \) which is continuous in the interval \ (\ left [(a, b) \ right]. \) Suppose that at some point \ ((x_0) \ in \ left [(a, b) \ right] \) the independent variable is incremented \ (\ Delta x. \) The function increment \ (\ Delta y, \) corresponding to such a change in argument \ (\ Delta x, \) is expressed by the formula \ [\ Delta y = \ Delta f \ left (((x_0)) \ right) = f \ left (((x_0) + \ Delta x) \ right) - f \ left (((x_0)) \ right) . \] For any differentiable function, the increment \ (\ Delta y \) can be represented as a sum of two terms: \ [\ Delta y = A \ Delta x + \ omicron \ left ((\ Delta x) \ right), \] where the first term (the so-called. main part increment) linearly depends on the increment \ (\ Delta x, \) and the second term has a higher order of smallness relative to \ (\ Delta x. \) Expression \ (A \ Delta x \) is called differential function and denoted by \ (dy \) or \ (df \ left (((x_0)) \ right). \)

Let's consider this idea of ​​splitting the function increment \ (\ Delta y \) into two parts using a simple example. Let a square with a side \ ((x_0) = 1 \, \ text (m) \, \) (figure \ (1 \)) be given. Its area is obviously \ [(S_0) = x_0 ^ 2 = 1 \, \ text (m) ^ 2. \] If the side of the square is increased by \ (\ Delta x = 1 \, \ text (cm), \ ) then the exact value of the area of ​​the enlarged square will be \ i.e. the area increment \ (\ Delta S \) equals \ [(\ Delta S = S - (S_0) = 1.0201 - 1 = 0.0201 \, \ text (m) ^ 2) = (201 \, \ text ( cm) ^ 2.) \] Now we represent this increment \ (\ Delta S \) as follows: \ [\ require (cancel) (\ Delta S = S - (S_0) = (\ left (((x_0) + \ Delta x) \ right) ^ 2) - x_0 ^ 2) = (\ cancel (x_0 ^ 2) + 2 (x_0) \ Delta x + (\ left ((\ Delta x) \ right) ^ 2) - \ cancel (x_0 ^ 2)) = (2 (x_0) \ Delta x + (\ left ((\ Delta x) \ right) ^ 2)) = (A \ Delta x + \ omicron \ left ((\ Delta x) \ right)) = (dy + o \ left ((\ Delta x) \ right).) \] So, the increment of the \ (\ Delta S \) function consists of the main part (the differential of the function), which is proportional to \ (\ Delta x \) and is equal to \ and a term of higher order of smallness, in turn equal to \ [\ omicron \ left ((\ Delta x) \ right) = (\ left ((\ Delta x) \ right) ^ 2) = (0.01 ^ 2) = 0.0001 \, \ text (m) ^ 2 = 1 \, \ text (cm) ^ 2. \] Together, both of these terms make up the total increment of the area of ​​the square, equal to \ (200 + 1 = 201 \, \ text (cm) ^ 2. \)

Note that in this example the coefficient \ (A \) is equal to the value of the derivative of the function \ (S \) at the point \ ((x_0): \) \ It turns out that for any differentiable function the following holds: theorem :

The coefficient \ (A \) of the main part of the increment of the function at the point \ ((x_0) \) is equal to the value of the derivative \ (f "\ left (((x_0)) \ right) \) at this point, that is, the increment \ ( \ Delta y \) is expressed by the formula \ [(\ Delta y = A \ Delta x + \ omicron \ left ((\ Delta x) \ right)) = (f "\ left (((x_0)) \ right) \ Delta x + \ omicron \ left ((\ Delta x) \ right).) \] Dividing both sides of this equality by \ (\ Delta x \ ne 0, \) we have \ [(\ frac ((\ Delta y)) ( (\ Delta x)) = A + \ frac ((\ omicron \ left ((\ Delta x) \ right))) ((\ Delta x))) = (f "\ left (((x_0)) \ right ) + \ frac ((\ omicron \ left ((\ Delta x) \ right))) ((\ Delta x)).) \] In the limit as \ (\ Delta x \ to 0 \), we obtain the value of the derivative at the point \ ((x_0): \) \ [(y "\ left (((x_0)) \ right) = \ lim \ limits _ (\ Delta x \ to 0) \ frac ((\ Delta y)) ((\ Delta x))) = (A = f "\ left (((x_0)) \ right).) \] Here we took into account that for a small value \ (\ omicron \ left ((\ Delta x) \ right) \) of a higher order of smallness than \ (\ Delta x, \) the limit is \ [\ lim \ limits _ (\ Delta x \ to 0) \ frac ((\ omicron \ left ((\ Delta x) \ right))) ( (\ Delta x)) = 0. \] If we assume that differential of the independent variable \ (dx \) is equal to its increment \ (\ Delta x: \) \ then from the relation \ it follows that \ i.e. the derivative of a function can be represented as the ratio of two differentials.

The geometric meaning of the differential of a function

Figure \ (2 \) schematically shows the breakdown of the function increment \ (\ Delta y \) into the main part \ (A \ Delta x \) (function differential) and a higher order term \ (\ omicron \ left ((\ Delta x ) \ right) \).

The tangent \ (MN \) drawn to the curve of the function \ (y = f \ left (x \ right) \) at the point \ (M \) is known to have an angle of inclination \ (\ alpha \), the tangent of which is equal to the derivative : \ [\ tan \ alpha = f "\ left (((x_0)) \ right). \] When the argument is changed to \ (\ Delta x \), the tangent is incremented \ (A \ Delta x. \) This is a linear increment formed by the tangent line is precisely the differential of the function.The rest of the total increment \ (\ Delta y \) (segment \ (N (M_1) \)) corresponds to a "nonlinear" addition with a higher order of smallness with respect to \ (\ Delta x \ ).

Differential properties

Let \ (u \) and \ (v \) be functions of the variable \ (x \). The differential has the following properties:

1. The constant coefficient can be taken outside the differential sign:

   \ (d \ left ((Cu) \ right) = Cdu \), where \ (C \) is a constant number.

2. Differential of the sum (difference) of functions:

   \ (d \ left ((u \ pm v) \ right) = du \ pm dv. \)

3. The differential of a constant value is equal to zero:

   \ (d \ left (C \ right) = 0. \)

4. The differential of the independent variable \ (x \) is equal to its increment:

   \ (dx = \ Delta x. \)

5. The differential of a linear function is equal to its increment:

   \ (d \ left ((ax + b) \ right) = \ Delta \ left ((ax + b) \ right) = a \ Delta x. \)

6. The differential of the product of two functions:

   \ (d \ left ((uv) \ right) = du \ cdot v + u \ cdot dv. \)

7. The quotient differential of two functions:

   \ (d \ left ((\ large \ frac (u) (v) \ normalsize) \ right) = \ large \ frac ((du \ cdot v - u \ cdot dv)) (((v ^ 2))) \ normalsize. \)

8. The differential of a function is equal to the product of the derivative and the differential of the argument:

   \ (dy = df \ left (x \ right) = f "\ left (x \ right) dx. \)

As you can see, the differential of the function \ (dy \) differs from the derivative only by the factor \ (dx \). For example, \ [(d \ left (((x ^ n)) \ right) = n (x ^ (n - 1)) dx,) \; \; (d \ left ((\ ln x) \ right) = \ frac ((dx)) (x),) \; \; (d \ left ((\ sin x) \ right) = \ cos x dx) \] and so on.

Form invariance of the differential

Consider the composition of two functions \ (y = f \ left (u \ right) \) and \ (u = g \ left (x \ right), \) i.e. complex function \ (y = f \ left ((g \ left (x \ right)) \ right). \) Its derivative is defined by the expression \ [(y "_x) = (y" _u) \ cdot (u "_x) , \] where the subscript denotes the variable over which differentiation is made.

The differential of the "outer" function \ (y = f \ left (u \ right) \) is written in the form \ The differential of the "inner" function \ (u = g \ left (x \ right) \) can be represented in a similar way: \ If you substitute \ (du \) into the previous formula, we get \ Since \ ((y "_x) = (y" _u) \ cdot (u "_x), \) then \ It can be seen that in the case of a complex function we got the same form expression for the differential of a function, as in the case of a "simple" function. This property is called the invariance of the form of the differential .