The differential function is the invariance of the form of the first differential. Properties of the first differential of a function

By definition, the differential (first differential) of a function is calculated by the formula
If – independent variable.

EXAMPLE.

Let us show that the form of the first differential remains unchanged (is invariant) even in the case when the argument of the function itself is a function, that is, for a complex function
.

Let
are differentiable, then by definition

Moreover, that is what needed to be proven.

EXAMPLES.

The proven invariance of the form of the first differential allows us to assume that
that is the derivative is equal to the ratio of the differential of the function to the differential of her argument, regardless of whether the argument is an independent variable or a function.

Differentiation of a function specified parametrically

Let If function
has on set the opposite, then
Then the equalities
defined on the set function specified parametrically, – parameter (intermediate variable).

EXAMPLE. Graph the function
.

The constructed curve is called cycloid(Fig. 25) and is the trajectory of a point on a circle of radius 1, which rolls without sliding along the OX axis.

COMMENT. Sometimes, but not always, a parameter can be eliminated from the parametric curve equations.

EXAMPLES.
are parametric equations of a circle, since, obviously,

–parametric equations of the ellipse, since

–parametric equations of a parabola

Let's find the derivative of a function defined parametrically:

The derivative of a function specified parametrically is also a function specified parametrically: .

DEFINITION. The second derivative of a function is the derivative of its first derivative.

Derivative the th order is the derivative of its derivative of order
.

Denote derivatives of the second and -th order like this:

From the definition of the second derivative and the rule of differentiation of a parametrically defined function it follows that
To calculate the third derivative, you need to represent the second derivative in the form
and use the resulting rule again. Higher order derivatives are calculated in a similar way.

EXAMPLE. Find the first and second order derivatives of the function

.

Basic theorems of differential calculus

THEOREM(Farm). Let the function
has at the point
extremum. If exists
, That

PROOF. Let
, for example, is the minimum point. By definition of a minimum point, there is a neighborhood of this point
, within which
, that is
– increment
at the point
. A-priory
Let's calculate one-sided derivatives at the point
:

by the theorem on passage to the limit in inequality,

because

, because
But according to the condition
exists, therefore the left derivative is equal to the right one, and this is only possible if

The assumption that
– the maximum point leads to the same thing.

Geometric meaning of the theorem:

THEOREM(Rolla). Let the function
continuous
, differentiable
And
then there is
such that

PROOF. Because
continuous
, then by Weierstrass’s second theorem it reaches at
their greatest
and the least
values ​​either at extremum points or at the ends of the segment.

1. Let
, Then

2. Let
Because
either
, or
is reached at the extremum point
, but according to Fermat's theorem
Q.E.D.

THEOREM(Lagrange). Let the function
continuous
and differentiable
, then there is
such that
.

Geometric meaning of the theorem:

Because
, then the secant is parallel to the tangent. Thus, the theorem states that there is a tangent parallel to the secant passing through points A and B.

PROOF. Through points A
and B
Let's draw a secant AB. Her equation
Consider the function

– the distance between the corresponding points on the graph and on the secant AB.

1.
continuous
as the difference of continuous functions.

2.
differentiable
as the difference of differentiable functions.

3.

Means,
satisfies the conditions of Rolle's theorem, so there exists
such that

The theorem has been proven.

COMMENT. The formula is called Lagrange's formula.

THEOREM(Cauchy). Let the functions
continuous
, differentiable
And
, then there is a point
such that
.

PROOF. Let's show that
. If
, then the function
would satisfy the conditions of Rolle's theorem, so there would be a point
such that
– contradiction to the condition. Means,
, and both sides of the formula are defined. Let's look at a helper function.

continuous
, differentiable
And
, that is
satisfies the conditions of Rolle's theorem. Then there is a point
, wherein
, But

Q.E.D.

The proven formula is called Cauchy formula.

L'Hopital's RULE(L'Hopital-Bernoulli theorem). Let the functions
continuous
, differentiable
,
And
. In addition, there is a finite or infinite
.

Then there is

PROOF. Since by condition
, then we define
at the point
, assuming
Then
will become continuous
. Let's show that

Let's pretend that
then there is
such that
, since the function
on
satisfies the conditions of Rolle's theorem. But according to the condition
– a contradiction. That's why

. Functions
satisfy the conditions of Cauchy's theorem on any interval
, which is contained in
. Let's write the Cauchy formula:

,
.

From here we have:
, because if
, That
.

Redesignating the variable in the last limit, we obtain the required:

NOTE 1. L'Hopital's rule remains valid even when
And
. It allows us to reveal not only the uncertainty of the type , but also the type :

.

NOTE 2. If, after applying L'Hopital's rule, the uncertainty is not revealed, then it should be applied again.

EXAMPLE.

COMMENT 3 . L'Hopital's rule is a universal way of revealing uncertainties, but there are limits that can be revealed by using only one of the previously studied particular techniques.

But obviously
, since the degree of the numerator is equal to the degree of the denominator, and the limit is equal to the ratio of the coefficients at the highest powers

The rule for differentiating a complex function will lead us to one remarkable and important property of the differential.

Let the functions be such that a complex function can be composed from them: . If derivatives exist, then - by rule V - there is also a derivative

Replacing, however, its derivative with expression (7) and noting that there is a differential of x as a function of t, we finally obtain:

i.e., let's return to the previous form of the differential!

Thus, we see that the shape of the differential can be preserved even if the old independent variable is replaced by a new one. We are always free to write the differential y in the form (5), whether x is an independent variable or not; the only difference is that if t is chosen as the independent variable, then it means not an arbitrary increment, but a differential of x as a function of This property is called the invariance of the form of the differential.

Since formula (5) directly yields formula (6), which expresses the derivative through differentials, the last formula remains valid no matter what independent variable (of course, the same in both cases) the said differentials are calculated.

Let, for example, so

Let us now put Then we will also have: It is easy to check that the formula

gives only another expression for the derivative calculated above.

This circumstance is especially convenient to use in cases where the dependence of y on x is not specified directly, but instead the dependence of both variables x and y on some third, auxiliary variable (called a parameter) is specified:

Assuming that both of these functions have derivatives and that for the first of them there is an inverse function that has a derivative, it is easy to see that then y also turns out to be a function of x:

for which there is also a derivative. The calculation of this derivative can be performed according to the above rule:

without restoring the direct dependence of y on x.

For example, if the derivative can be determined as done above, without using the dependence at all.

If we consider x and y as rectangular coordinates of a point on the plane, then equations (8) assign each value of the parameter t to a certain point, which, with a change in t, describes a curve on the plane. Equations (8) are called parametric equations of this curve.

In the case of a parametric definition of a curve, formula (10) allows you to directly set the slope of the tangent using equations (8), without proceeding to specifying the curve using equation (9); exactly,

Comment. The ability to express the derivative through differentials taken with respect to any variable, in particular, leads to the fact that the formulas

expressing in Leibniz notation the rules for differentiating an inverse function and a complex function, become simple algebraic identities (since all the differentials here can be taken with respect to the same variable). One should not think, however, that this gives a new conclusion to the above-mentioned formulas: first of all, the existence of left derivatives was not proved here, the main thing is that we essentially used the invariance of the form of the differential, which itself is a consequence of Rule V.

If a differentiable function of independent variables and its total differential dz is equal to Let Now Assume that at the point ((,?/) the functions »?) and r)) have continuous partial derivatives with respect to (and rf, and at the corresponding point (x, y ) partial derivatives exist and are continuous, and as a result the function r = f(x, y) is differentiable at this point. Under these conditions, the function has derivatives at the point 17) Differential of a complex function Invariance of the form of a differential Implicit functions Tangent plane and normal to the surface Tangent plane of the surface. Geometric meaning of the total differential Normal to the surface As can be seen from formulas (2), u and u are continuous at the point ((,*?). Therefore, the function at the point is differentiable; according to the formula of the total differential for a function of independent variables £ and m], we have Replacing on the right side of equalities (3) u and u their expressions from formulas (2), we obtain either that, according to the condition, the functions at the point ((,17) have continuous partial derivatives, then they are differentiable at this point and From relations (4) and (5) we obtain that Comparison of formulas (1) and (6) shows that the total differential of the function z = /(z, y) is expressed by a formula of the same form as in the case when the arguments x and y of the function /(z, y) are independent variables, and in the case when these arguments are, in turn, functions of some variables. Thus, the total differential of a function of several variables has the property of form invariance. Comment. From the invariance of the form of the total differential it follows: if xnx and y are differentiable functions of any finite number of variables, then the formula remains valid. Let us have the equation where is a function of two variables defined in some domain G on the xOy plane. If for each value x from a certain interval (xo - 0, xo + ^o) there is exactly one value y, which together with x satisfies equation (1), then this determines the function y = y(x), for which the equality is written identically along x in the specified interval. In this case, equation (1) is said to define y as an implicit function of x. In other words, a function specified by an equation that is not resolved with respect to y is called an implicit function,” it becomes explicit if the dependence of y on x is given directly. Examples: 1. The equation defines the value y on the entire OcW рх as a single-valued function of x: 2. By the equation the quantity y is defined as a single-valued function of x. Let us illustrate this statement. The equation is satisfied by a pair of values ​​x = 0, y = 0. We will consider * a parameter and consider the functions. The question of whether, for the chosen xo, there is a corresponding unique value of O is such that the pair (satisfies equation (2) comes down to intersecting the x ay curves and a single point. Let us construct their graphs on the xOy plane (Fig. 11) The curve » = x + c sin y, where x is considered as a parameter, is obtained by parallel translation along the Ox axis and the curve z = z sin y. It is geometrically obvious that for any x the curves x = y and z = t + c $1py have a unique one. » intersection point, the ordinator of which is a function of x, determined by equation (2) implicitly. This dependence is not expressed through elementary functions. 3. The equation does not determine the real function of x in the same argument. In a sense, we can talk about implicit functions of several variables. The following theorem gives sufficient conditions for the unique solvability of the equation = 0 (1) in some neighborhood of a given point (®o> 0). Theorem 8 (the existence of an implicit function) Let the following conditions be satisfied: 1) the function is defined and continuous in a certain rectangle with center at a point at the point the function y) turns into n\l, 3) in the rectangle D there exist and continuous partial derivatives 4) Y) When any sufficiently ma/sueo positive number e there is a neighborhood of this neighborhood there is a single continuous function y = f(x) (Fig. 12), which takes the value), satisfies the equation \y - yol and turns equation (1) into the identity: This function is continuously differentiable in a neighborhood of the point Xq, and Let us derive formula (3) for the derivative of the implicit function, considering the existence of this derivative to be proven. Let y = f(x) be the implicit differentiable function defined by equation (1). Then in the interval) there is an identity Differential of a complex function Invariance of the form of a differential Implicit functions Tangent plane and normal to a surface Tangent plane of a surface Geometric meaning of a complete differential Normal to a surface due to it in this interval According to the rule of differentiation of a complex function, we have Unique in the sense that any point (x , y), lying on the curve belonging to the neighborhood of the point (xo, yo)” has coordinates related by the equation. Hence, with y = f(x) we obtain that and, therefore, Example. Find j* from the function y = y(x), defined by the equation In this case From here, by virtue of formula (3) Remark. The theorem will provide conditions for the existence of a single implicit function whose graph passes through a given point (xo, oo). sufficient, but not necessary. As a matter of fact, consider the equation Here has continuous partial derivatives equal to zero at the point 0(0,0). However, this equation has a unique solution equal to zero at Problem. Let an equation be given - a single-valued function that satisfies equation (G). 1) How many single-valued functions (2") satisfy the equation (!")? 2) How many single-valued continuous functions satisfy the equation (!")? 3) How many single-valued differentiable functions satisfy the equation (!")? 4) How many single-valued continuous functions satisfy “equation (1”), even if they are small enough? An existence theorem similar to Theorem 8 also holds in the case of an implicit function z - z(x, y) of two variables defined by the equation Theorem 9. Let the following conditions be satisfied d) the function & is defined and continuous in the domain D in the domain D there exist and continuous partial derivatives Then for any sufficiently small e > 0 there is a neighborhood Γ2 of the point (®o»Yo)/ in which there is a unique continuous function z - /(x, y), taking a value at x = x0, y = y0, satisfying the condition and reversing equation (4) into the identity: In this case, the function in the domain Q has continuous partial derivatives and GG Let us find expressions for these derivatives. Let the equation define z as a single-valued and differentiable function z = /(x, y) of independent variables xnu. If we substitute the function f(x, y) into this equation instead of z, we obtain the identity Consequently, the total partial derivatives with respect to x and y of the function y, z), where z = /(z, y), must also be equal to zero. By differentiating, we find where These formulas give expressions for the partial derivatives of the implicit function of two independent variables. Example. Find the partial derivatives of the function x(r,y) given by equation 4. From this we have §11. Tangent plane and normal to the surface 11.1. Preliminary information Let us have a surface S defined by the equation Defined*. A point M(x, y, z) of surface (1) is called an ordinary point of this surface if at point M all three derivatives exist and are continuous, and at least one of them is nonzero. If at point My, z) of surface (1) all three derivatives are equal to zero or at least one of these derivatives does not exist, then point M is called a singular point of the surface. Example. Consider a circular cone (Fig. 13). Here the only special subtle point is the origin of coordinates 0(0,0,0): at this point the partial derivatives simultaneously vanish. Rice. 13 Consider a spatial curve L defined by parametric equations. Let the functions have continuous derivatives in the interval. Let us exclude from consideration the singular points of the curve at which Let be an ordinary point of the curve L, determined by the value of the to parameter. Then is the tangent vector to the curve at the point. Tangent plane of a surface Let the surface 5 be given by the equation. Take an ordinary point P on the surface S and draw through it some curve L lying on the surface and given by parametric equations. Assume that the functions £(*), "/(0" C(0) have continuous derivatives , nowhere on (a)p) which simultaneously vanish. By definition, the tangent of the curve L at point P is called tangent to the surface 5 at this point. If expressions (2) are substituted into equation (1), then since the curve L lies. on the surface S, equation (1) turns into an identity with respect to t: Differentiating this identity with respect to t, using the rule for differentiating a complex function, we obtain The expression on the left side of (3) is the scalar product of two vectors: At point P, the vector z is directed tangent to the curve L at this point (Fig. 14). As for the vector n, it depends only on the coordinates of this point and the type of function ^"(x, y, z) and does not depend on the type of the curve passing through the point P. Since P - ordinary point of the surface 5, then the length of the vector n is different from zero. The fact that the scalar product means that the vector r tangent to the curve L at point P is perpendicular to the vector n at this point (Fig. 14). These arguments remain valid for any curve passing through point P and lying on the surface S. Consequently, any tangent line to the surface 5 at point P is perpendicular to the vector n, and, therefore, all these lines lie in the same plane, also perpendicular to the vector n . Definition. The plane in which all tangent lines to surface 5 passing through a given ordinary point P G 5 are located is called the tangent plane of the surface at point P (Fig. 15). Vector Differential of a complex function Invariance of the form of the differential Implicit functions Tangent plane and normal to the surface Tangent plane of the surface Geometric meaning of the complete differential The normal to the surface is the normal vector of the tangent plane to the surface at point P. From here we immediately obtain the equation of the tangent plane to the surface ZG (at the ordinary point P0 (®o, Uo" of this surface: If surface 5 is given by an equation, then by writing this equation in the form we also obtain the equation of the tangent plane at the point, it will look like this 11. 3. Geometric meaning of the total differential If we put it in formula (7), then it will take the form The right side of (8) represents the total differential of the function z at the point M0(x0) yо) on the plane xOy> so that Thus, the total differential of the function z = /(x, y) of two independent variables x and y at point M0, corresponding to the increments Dx and Du of the variables and y, is equal to the increment z - z0 applicates z of the point of the tangent plane of the surface 5 at the point Z>(xo» Uo» /(, Uo)) WHEN moving from point M0(xo, Uo) to point - 11.4. Surface Normal Definition. The straight line passing through the point Po(xo, y0, r0) of the surface perpendicular to the tangent plane to the surface at the point Po is called the normal to the surface at the point Pq. Vector)L is the directing vector of the normal, and its equations have the form If surface 5 is given by an equation, then the equations of the normal at the point) look like this: at the point Here At the point (0,0) these derivatives are equal to zero: and the equation of the tangent plane at the point 0 (0,0,0) takes the following form: (xOy plane). Normal equations

The formula for the differential function has the form

where is the differential of the independent variable.

Let now be given a complex (differentiable) function , where,.Then using the formula for the derivative of a complex function we find

because .

So, , i.e. The differential formula has the same form for the independent variable and for the intermediate argument, which is a differentiable function of.

This property is usually called the property invariance of a formula or form of a differential. Note that the derivative does not have this property.

Relationship between continuity and differentiability.

Theorem (a necessary condition for the differentiability of a function). If a function is differentiable at a point, then it is continuous at that point.

Proof. Let the function y=f(x) differentiable at the point X 0 . At this point we give the argument an increment  X. The function will be incremented  at. Let's find it.

Hence, y=f(x) continuous at a point X 0 .

Consequence. If X 0 is the discontinuity point of the function, then the function at it is not differentiable.

The converse of the theorem is not true. Continuity does not imply differentiability.

Differential. Geometric meaning. Application of differential to approximate calculations.

Definition

Function differential is called the linear relative part of the increment of the function. It is designated kakili. Thus:

Comment

The differential of a function makes up the bulk of its increment.

Comment

Along with the concept of a function differential, the concept of an argument differential is introduced. A-priory argument differential is the increment of the argument:

Comment

The formula for the differential of a function can be written as:

From here we get that

So, this means that the derivative can be represented as an ordinary fraction - the ratio of the differentials of a function and an argument.

Geometric meaning of differential

The differential of a function at a point is equal to the ordinate increment of the tangent drawn to the graph of the function at this point, corresponding to the increment of the argument.

Basic rules of differentiation. Derivative of a constant, derivative of a sum.

Let the functions have derivatives at a point. Then

1. Constant can be taken out of the derivative sign.

5. Differential constant equal to zero.

2. Derivative of sum/difference.

The derivative of the sum/difference of two functions is equal to the sum/difference of the derivatives of each function.

Basic rules of differentiation. Derivative of the product.

3. Derivative of the product.

Basic rules of differentiation. Derivative of a complex and inverse function.

5. Derivative of a complex function.

The derivative of a complex function is equal to the derivative of this function with respect to the intermediate argument, multiplied by the derivative of the intermediate argument with respect to the main argument.

And they have derivatives at points, respectively. Then

Theorem

(About the derivative of the inverse function)

If a function is continuous and strictly monotone in some neighborhood of a point and differentiable at this point, then the inverse function has a derivative at the point, and .

Differentiation formulas. Derivative of an exponential function.