Let x be an arbitrary point lying in some neighborhood of a fixed point x 0 . the difference x - x 0 is usually called the increment of the independent variable (or the increment of the argument) at the point x 0 and is denoted by Δx. Thus,
Δx \u003d x - x 0,
whence it follows that
Function Increment − difference between two function values.
Let the function at = f(x), defined when the value of the argument is equal to X 0 . Let's increment D X, ᴛ.ᴇ. consider the value of the argument ͵ equal to x 0+D X. Assume that this argument value is also within the scope of this function. Then the difference D y = f(x 0+D X) – f(x0) is called the increment of a function. Function increment f(x) at the point x is a function usually denoted Δ x f on the new variable Δ x defined as
Δ x f(Δ x) = f(x + Δ x) − f(x).
Find the increment of the argument and the increment of the function at the point x 0 if
Example 2. Find the increment of the function f (x) \u003d x 2 if x \u003d 1, ∆x \u003d 0.1
Solution: f (x) \u003d x 2, f (x + ∆x) \u003d (x + ∆x) 2
Find the increment of the function ∆f = f(x+∆x) - f(x) = (x+∆x) 2 - x 2 = x 2 +2x*∆x+∆x 2 - x 2 = 2x*∆x + ∆x 2 /
Substitute the values x=1 and ∆x= 0.1, we get ∆f = 2*1*0.1 + (0.1) 2 = 0.2+0.01 = 0.21
Find the increment of the argument and the increment of the function at points x 0
2.f(x) \u003d 2x 3. x 0 \u003d 3 x \u003d 2.4
3. f(x) \u003d 2x 2 +2 x 0 \u003d 1 x \u003d 0.8
4. f(x) \u003d 3x + 4 x 0 \u003d 4 x \u003d 3.8
Definition: Derivative It is customary to call a function at a point the limit (if it exists and is finite) of the ratio of the increment of the function to the increment of the argument, provided that the latter tends to zero.
The following notation for the derivative is most commonly used:
Thus,
Finding the derivative is called differentiation . Introduced definition of a differentiable function: A function f that has a derivative at every point of some interval is called differentiable on this interval.
Let a function be defined in some neighborhood of the point. It is customary to call the derivative of a function such a number that the function in the neighborhood U(x 0) can be represented as
f(x 0 + h) = f(x 0) + Ah + o(h)
if exists.
Definition of the derivative of a function at a point.
Let the function f(x) defined on the interval (a;b), and are the points of this interval.
Definition. Derivative function f(x) at a point, it is customary to call the limit of the ratio of the increment of a function to the increment of the argument at . Designated .
When the last limit takes on a specific final value, then one speaks of the existence final derivative at a point. If the limit is infinite, then we say that derivative is infinite at a given point. If the limit does not exist, then the derivative of the function does not exist at this point.
Function f(x) is said to be differentiable at a point when it has a finite derivative at it.
In case the function f(x) is differentiable at every point of some interval (a;b), then the function is called differentiable on this interval. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, any point x from the gap (a;b) we can associate the value of the derivative of the function at this point, that is, we have the opportunity to define a new function, which is called the derivative of the function f(x) on the interval (a;b).
The operation of finding the derivative is called differentiation.
Not always in life we are interested in the exact values of any quantities. Sometimes it is interesting to know the change in this value, for example, average speed bus, the ratio of the amount of movement to the period of time, etc. To compare the value of a function at some point with the values of the same function at other points, it is convenient to use concepts such as "function increment" and "argument increment".
The concepts of "function increment" and "argument increment"
Suppose x is some arbitrary point that lies in some neighborhood of the point x0. The increment of the argument at the point x0 is the difference x-x0. The increment is denoted as follows: ∆x.
- ∆x=x-x0.
Sometimes this value is also called the increment of the independent variable at the point x0. It follows from the formula: x = x0 + ∆x. In such cases, it is said that the initial value of the independent variable x0 has received an increment ∆x.
If we change the argument, then the value of the function will also change.
- f(x) - f(x0) = f(x0 + ∆х) - f(x0).
The increment of the function f at the point x0, the corresponding increment ∆x is the difference f(x0 + ∆x) - f(x0). The increment of a function is denoted as ∆f. Thus we get, by definition:
- ∆f= f(x0 + ∆x) - f(x0).
Sometimes, ∆f is also called the increment of the dependent variable and ∆y is used to denote it if the function was, for example, y=f(x).
Geometric sense of increment
Look at the next picture.
As you can see, the increment shows the change in the ordinate and abscissa of the point. And the ratio of the increment of the function to the increment of the argument determines the angle of inclination of the secant passing through the initial and final positions of the point.
Consider examples of function and argument increment
Example 1 Find the increment of the argument ∆x and the increment of the function ∆f at the point x0 if f(x) = x 2 , x0=2 a) x=1.9 b) x =2.1
Let's use the formulas above:
a) ∆х=х-х0 = 1.9 - 2 = -0.1;
- ∆f=f(1.9) - f(2) = 1.9 2 - 2 2 = -0.39;
b) ∆x=x-x0=2.1-2=0.1;
- ∆f=f(2.1) - f(2) = 2.1 2 - 2 2 = 0.41.
Example 2 Calculate the increment ∆f for the function f(x) = 1/x at the point x0 if the increment of the argument is equal to ∆x.
Again, we use the formulas obtained above.
- ∆f = f(x0 + ∆x) - f(x0) =1/(x0-∆x) - 1/x0 = (x0 - (x0+∆x))/(x0*(x0+∆x)) = -∆x/((x0*(x0+∆x)).
Let X– argument (independent variable); y=y(x)- function.
Take a fixed value of the argument x=x 0 and calculate the value of the function y 0 =y(x 0 ) . We now arbitrarily set increment (change) of the argument and denote it X ( X can be of any sign).
Incremental argument is a point X 0 + X. Suppose it also contains a function value y=y(x 0 + X)(see picture).
Thus, with an arbitrary change in the value of the argument, a change in the function is obtained, which is called increment function values:
and is not arbitrary, but depends on the type of function and quantity 
.
Argument and function increments can be final, i.e. expressed as constant numbers, in which case they are sometimes called finite differences.
In economics, finite increments are considered quite often. For example, the table shows data on the length of the railway network of a certain state. Obviously, the network length increment is calculated by subtracting the previous value from the next.
We will consider the length of the railway network as a function, the argument of which will be time (years).
|
Railway length as of December 31, thousand km |
Increment |
Average annual growth |
|
In itself, the increment of the function (in this case, the length of the railway network) poorly characterizes the change in the function. In our example, from the fact that 2,5>0,9 cannot be concluded that the network grew faster in 2000-2003 years than in 2004 g., because the increment 2,5 refers to a three-year period, and 0,9 - in just one year. Therefore, it is quite natural that the increment of the function leads to a unit change in the argument. The argument increment here is periods: 1996-1993=3; 2000-1996=4; 2003-2000=3; 2004-2003=1 .
We get what is called in the economic literature average annual growth.
You can avoid the operation of casting the increment to the unit of change of the argument, if you take the values of the function for the values of the argument that differ by one, which is not always possible.
In mathematical analysis, in particular, in differential calculus, infinitesimal (IM) increments of an argument and a function are considered.
Differentiation of a function of one variable (derivative and differential) Derivative of a function
Argument and function increments at point X 0 can be considered as comparable infinitesimal quantities (see topic 4, comparison of BM), i.e. BM of the same order.
Then their ratio will have a finite limit, which is defined as the derivative of the function in t X 0 .
Limit of ratio of function increment to BM argument increment at a point x=x 0 called derivative functions at this point.
The symbolic designation of the derivative with a stroke (or rather, the Roman numeral I) was introduced by Newton. You can also use a subscript that shows which variable the derivative is calculated from, for example, 
. Another notation proposed by the founder of the calculus of derivatives, the German mathematician Leibniz, is also widely used: 
. You will learn more about the origin of this designation in the section Function differential and argument differential.
This number evaluates speed changing the function passing through the point 
.
Let's install geometric meaning derivative of a function at a point. To this end, we construct a graph of the function y=y(x) and mark on it the points that determine the change y(x) in the interim 
Tangent to the graph of a function at a point M 0
we will consider the limiting position of the secant M 0
M given that 
(dot M slides along the graph of the function to a point M 0
).
Consider 
. Obviously, 
.
If the point M rush along the graph of the function towards the point M 0
, then the value 
will tend to a certain limit, which we denote 
. Wherein.
Limit angle
coincides with the angle of inclination of the tangent drawn to the graph of the function, incl. M 0
, so the derivative 
is numerically equal to tangent slope
at the specified point.
-
geometric meaning of the derivative of a function at a point.
Thus, one can write down the equations of the tangent and normal ( normal is a line perpendicular to the tangent) to the graph of the function at some point X 0 :
Tangent - .
Normal - 
.
Of interest are the cases when these lines are located horizontally or vertically (see topic 3, special cases of the position of a line on a plane). Then,
If 
;
If 
.
The definition of a derivative is called differentiation functions.
If the function at the point X 0 has a finite derivative, it is called differentiable at this point. A function that is differentiable at all points of some interval is called differentiable on this interval.
Theorem . If the function y=y(x) differentiable in t. X 0 , then it is continuous at this point.
Thus, continuity is a necessary (but not sufficient) condition for the function to be differentiable.
Definition 1
If for each pair $(x,y)$ of values of two independent variables from some domain a certain value of $z$ is assigned, then $z$ is said to be a function of two variables $(x,y)$. Notation: $z=f(x,y)$.
In a relationship functions$z=f(x,y)$ consider the concepts of general (total) and partial increments of a function.
Let a function $z=f(x,y)$ of two independent variables $(x,y)$ be given.
Remark 1
Since the variables $(x,y)$ are independent, one of them can change while the other remains constant.
Let's give the variable $x$ an increment $\Delta x$, while keeping the value of the variable $y$ unchanged.
Then the function $z=f(x,y)$ will receive an increment, which will be called the partial increment of the function $z=f(x,y)$ with respect to the variable $x$. Designation:
Similarly, we give the variable $y$ an increment $\Delta y$, while keeping the value of the variable $x$ unchanged.
Then the function $z=f(x,y)$ will receive an increment, which will be called the partial increment of the function $z=f(x,y)$ with respect to the variable $y$. Designation:
If the argument $x$ is incremented by $\Delta x$, and the argument $y$ is incremented by $\Delta y$, then we get full increment given function $z=f(x,y)$. Designation:
Thus, we have:
$\Delta _(x) z=f(x+\Delta x,y)-f(x,y)$ - partial increment of the function $z=f(x,y)$ with respect to $x$;
$\Delta _(y) z=f(x,y+\Delta y)-f(x,y)$ - partial increment of the function $z=f(x,y)$ with respect to $y$;
$\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)$ - total increment of the function $z=f(x,y)$.
Example 1
Solution:
$\Delta _(x) z=x+\Delta x+y$ - partial increment of the function $z=f(x,y)$ with respect to $x$;
$\Delta _(y) z=x+y+\Delta y$ - partial increment of the function $z=f(x,y)$ with respect to $y$.
$\Delta z=x+\Delta x+y+\Delta y$ - total increment of the function $z=f(x,y)$.
Example 2
Calculate the partial and total increments of the function $z=xy$ at the point $(1;2)$ for $\Delta x=0.1;\, \, \Delta y=0.1$.
Solution:
By definition of a private increment, we find:
$\Delta _(x) z=(x+\Delta x)\cdot y$ - partial increment of the function $z=f(x,y)$ with respect to $x$
$\Delta _(y) z=x\cdot (y+\Delta y)$ - partial increment of the function $z=f(x,y)$ with respect to $y$;
By the definition of the total increment, we find:
$\Delta z=(x+\Delta x)\cdot (y+\Delta y)$ - total increment of the function $z=f(x,y)$.
Hence,
\[\Delta _(x) z=(1+0.1)\cdot 2=2.2\] \[\Delta _(y) z=1\cdot (2+0.1)=2.1\] \[\Delta z=(1+0.1)\cdot (2+0.1)=1.1\cdot 2.1=2.31.\]
Remark 2
The total increment of the given function $z=f(x,y)$ is not equal to the sum of its partial increments $\Delta _(x) z$ and $\Delta _(y) z$. Mathematical notation: $\Delta z\ne \Delta _(x) z+\Delta _(y) z$.
Example 3
Check statement remarks for a function
Solution:
$\Delta _(x) z=x+\Delta x+y$; $\Delta _(y) z=x+y+\Delta y$; $\Delta z=x+\Delta x+y+\Delta y$ (obtained in example 1)
Find the sum of partial increments of the given function $z=f(x,y)$
\[\Delta _(x) z+\Delta _(y) z=x+\Delta x+y+(x+y+\Delta y)=2\cdot (x+y)+\Delta x+\Delta y.\]
\[\Delta _(x) z+\Delta _(y) z\ne \Delta z.\]
Definition 2
If for each triple $(x,y,z)$ of values of three independent variables from some domain, a certain value of $w$ is assigned, then $w$ is said to be a function of three variables $(x,y,z)$ in the given domain.
Notation: $w=f(x,y,z)$.
Definition 3
If for each collection $(x,y,z,...,t)$ of values of independent variables from some domain, a certain value of $w$ is assigned, then $w$ is said to be a function of the variables $(x,y,z,...,t)$ in the given domain.
Notation: $w=f(x,y,z,...,t)$.
For a function of three or more variables, in the same way as for a function of two variables, partial increments are determined for each of the variables:
$\Delta _(z) w=f(x,y,z+\Delta z)-f(x,y,z)$ - partial increment of the function $w=f(x,y,z,...,t)$ with respect to $z$;
$\Delta _(t) w=f(x,y,z,...,t+\Delta t)-f(x,y,z,...,t)$ - partial increment of the function $w=f(x,y,z,...,t)$ with respect to $t$.
Example 4
Write partial and total increments of a function
Solution:
By definition of a private increment, we find:
$\Delta _(x) w=((x+\Delta x)+y)\cdot z$ - partial increment of the function $w=f(x,y,z)$ with respect to $x$
$\Delta _(y) w=(x+(y+\Delta y))\cdot z$ - partial increment of the function $w=f(x,y,z)$ with respect to $y$;
$\Delta _(z) w=(x+y)\cdot (z+\Delta z)$ - partial increment of the function $w=f(x,y,z)$ with respect to $z$;
By the definition of the total increment, we find:
$\Delta w=((x+\Delta x)+(y+\Delta y))\cdot (z+\Delta z)$ - total increment of the function $w=f(x,y,z)$.
Example 5
Calculate the partial and total increments of the function $w=xyz$ at the point $(1;2;1)$ for $\Delta x=0.1;\, \, \Delta y=0.1;\, \, \Delta z=0.1$.
Solution:
By definition of a private increment, we find:
$\Delta _(x) w=(x+\Delta x)\cdot y\cdot z$ - partial increment of the function $w=f(x,y,z)$ with respect to $x$
$\Delta _(y) w=x\cdot (y+\Delta y)\cdot z$ - partial increment of the function $w=f(x,y,z)$ with respect to $y$;
$\Delta _(z) w=x\cdot y\cdot (z+\Delta z)$ - partial increment of the function $w=f(x,y,z)$ with respect to $z$;
By the definition of the total increment, we find:
$\Delta w=(x+\Delta x)\cdot (y+\Delta y)\cdot (z+\Delta z)$ - total increment of the function $w=f(x,y,z)$.
Hence,
\[\Delta _(x) w=(1+0,1)\cdot 2\cdot 1=2,2\] \[\Delta _(y) w=1\cdot (2+0,1)\cdot 1=2,1\] \[\Delta _(y) w=1\cdot 2\cdot (1+0,1)=2,2\] \[\Delta z=( 1+0.1)\cdot (2+0.1)\cdot (1+0.1)=1.1\cdot 2.1\cdot 1.1=2.541.\]
WITH geometric point From the point of view, the total increment of the function $z=f(x,y)$ (by definition $\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)$) is equal to the increment of the applicate of the graph of the function $z=f(x,y)$ when passing from the point $M(x,y)$ to the point $M_(1) (x+\Delta x,y+\Delta y)$ (Fig. 1).
Picture 1.