in medical and biological physics
LECTURE No. 1
DERIVATIVE AND DIFFERENTIAL FUNCTIONS.
PARTIAL DERIVATIVES.
1. The concept of derivative, its mechanical and geometric meaning.
A ) Increment of argument and function.
Let a function y=f(x) be given, where x is the value of the argument from the domain of definition of the function. If you select two values of the argument x o and x from a certain interval of the domain of definition of the function, then the difference between the two values of the argument is called the increment of the argument: x - x o =∆x.
The value of the argument x can be determined through x 0 and its increment: x = x o + ∆x.
The difference between two function values is called the function increment: ∆y =∆f = f(x o +∆x) – f(x o).
The increment of an argument and a function can be represented graphically (Fig. 1). Argument increment and function increment can be either positive or negative. As follows from Fig. 1, geometrically, the increment of the argument ∆х is represented by the increment of the abscissa, and the increment of the function ∆у by the increment of the ordinate. The function increment should be calculated in the following order:
we give the argument an increment ∆x and get the value – x+Δx;
2) find the value of the function for the value of the argument (x+∆x) – f(x+∆x);
3) find the increment of the function ∆f=f(x + ∆x) - f(x).
Example: Determine the increment of the function y=x 2 if the argument changed from x o =1 to x=3. For point x o the value of the function f(x o) = x² o; for the point (x o +∆x) the value of the function f(x o +∆x) = (x o +∆x) 2 = x² o +2x o ∆x+∆x 2, from where ∆f = f(x o + ∆x)–f(x o) = (x o +∆x) 2 –x² o = x² o +2x o ∆x+∆x 2 –x² o = 2x o ∆x+∆x 2; ∆f = 2x o ∆x+∆x 2 ;
∆х = 3–1 = 2; ∆f =2·1·2+4 = 8.b)
Problems leading to the concept of derivative. Definition of derivative, its physical meaning.
The concept of increment of argument and function is necessary to introduce the concept of derivative, which historically arose based on the need to determine the speed of certain processes.
Let's consider how you can determine the speed of rectilinear motion. Let the body move rectilinearly according to the law: ∆S= ·∆t. For uniform motion:= ∆S/∆t. For alternating motion, the value ∆Ѕ/∆t determines the value avg. , i.e. avg. =∆S/∆t.But does not make it possible to reflect the features of body movement and give an idea of the true speed at time t. When the time period decreases, i.e. at ∆t→0 the average speed tends to its limit – the instantaneous speed:
instant = 
avg. = 
∆S/∆t.
The instantaneous rate of a chemical reaction is determined in the same way:
instant = 
avg. = 
∆х/∆t,
where x is the amount of substance formed during a chemical reaction during time t. Similar problems of determining the speed of various processes led to the introduction in mathematics of the concept of a derivative function.
Let it be given continuous function f(x), defined on the interval ]a, in[ie its increment ∆f=f(x+∆x)–f(x).Relation 
is a function of ∆x and expresses the average rate of change of the function.
Ratio limit 
, when ∆х→0, provided that this limit exists, is called the derivative of the function :
y" x = 
.
The derivative is denoted: 
– (Yigree stroke by X);f "
(x) – (eff prime on x) ;
y" – (Greek stroke); dy/dх –
(de igrek by de x);
- (Greek with a dot).
Based on the definition of the derivative, we can say that the instantaneous speed of rectilinear motion is the time derivative of the path:
instant = S" t = f " (t).
Thus, we can conclude that the derivative of a function with respect to the argument x is the instantaneous rate of change of the function f(x):
y" x =f " (x)= instant.
This is the physical meaning of the derivative. The process of finding the derivative is called differentiation, so the expression “differentiate a function” is equivalent to the expression “find the derivative of a function.”
V)Geometric meaning of derivative.
P 
the derivative of the function y = f(x) has a simple geometric meaning associated with the concept of a tangent to a curved line at some point M. At the same time, tangent, i.e. a straight line is analytically expressed as y = kx = tan· x, where
–
the angle of inclination of the tangent (straight line) to the X axis. Let us imagine a continuous curve as a function y = f(x), take a point M1 on the curve and a point M1 close to it and draw a secant through them. Its slope to sec =tg β = 
.If we bring point M 1 closer to M, then the increment in argument ∆х
will tend to zero, and the secant at β=α will take the position of a tangent. From Fig. 2 it follows: tgα = 
tgβ = 
=y" x. But tgα is equal to the slope of the tangent to the graph of the function:
k = tgα = 
=y" x = f "
(X). So, the angular coefficient of a tangent to the graph of a function at a given point is equal to the value of its derivative at the point of tangency. This is the geometric meaning of the derivative.
G)General rule for finding the derivative.
Based on the definition of the derivative, the process of differentiating a function can be represented as follows:
f(x+∆x) = f(x)+∆f;
find the increment of the function: ∆f= f(x + ∆x) - f(x);
form the ratio of the increment of the function to the increment of the argument:
;
Example: f(x)=x 2 ; " f
(x)=?.
However, as can be seen even from this simple example, the use of the specified sequence when taking derivatives is a labor-intensive and complex process. Therefore, for various functions, general differentiation formulas are introduced, which are presented in the form of a table of “Basic formulas for differentiation of functions.”1. argument increment and function increment. 
Let the function be given. Let's take two argument values: initial 
and modified, which is usually denoted 
, Where - the amount by which the argument changes when moving from the first value to the second, it is called
argument increment. 
The argument values and correspond to specific function values: initial 
and changed 
, magnitude , by which the value of the function changes when the argument changes by value, is called
function increment.
2. the concept of the limit of a function at a point. 
Number 
called the limit of the function 
with tending to 
, if for any number 
there is such a number 
that in front of everyone 
, satisfying the inequality 
.
, the inequality will be satisfied 
.
Second definition: A number is called the limit of a function as it tends to , if for any number there is a neighborhood of the point such that for any of this neighborhood . Designated 3. infinitely large and infinitesimal functions at a point. An infinitesimal function at a point is a function whose limit, when it approaches a given point, is zero. Endlessly great function
at a point - a function whose limit when it tends to a given point is equal to infinity.
4. main theorems about limits and consequences from them (without proof). 
consequence: the constant factor can be taken beyond the limit sign: 
If the sequences and
converge and the limit of the sequence is nonzero, then
consequence: the constant factor can be taken beyond the limit sign. 
11. if there are limits to functions 
And
and the limit of the function is non-zero,
.
then there is also a limit of their ratio, equal to the ratio of the limits of the functions and : 
12. if 
, That
, the converse is also true. 
13. Theorem on the limit of an intermediate sequence. If the sequences 
11. if there are limits to functions 
converging, and 
That
5. limit of a function at infinity. 
corresponds to a sequence of values tending to the number A.
6. limits number sequence.
2. the concept of the limit of a function at a point. A is called the limit of a number sequence if for any positive number 
there is a natural number N such that for all n>
N inequality holds 
.
Symbolically this is defined as follows: 
fair .
The fact that the number A is the limit of the sequence, denoted as follows:
.
7.number "e". natural logarithms.
2. the concept of the limit of a function at a point. "e"
represents the limit of the number sequence, n-
th member of which 
, i.e.
.
Natural logarithm – logarithm with a base e.
natural logarithms are denoted 
without specifying a reason. 
2. the concept of the limit of a function at a point. 
allows you to go from the decimal logarithm to the natural one and back.
, it is called the modulus of transition from natural logarithms to decimal ones.
8. wonderful limits 
,
.
The first remarkable limit:
thus at 
by the intermediate sequence limit theorem
second remarkable limit:
.
To prove the existence of a limit 
use the lemma: for any real number 
And 
inequality is true 
(2) (at 
or 
inequality turns into equality.)
Sequence (1) can be written as follows:
.
Now consider an auxiliary sequence with a common term 
Let's make sure that it decreases and is bounded below: 
If 
, then the sequence decreases. If 
, then the sequence is bounded below. Let's show this:
due to equality (2)
i.e. 
or 
. That is, the sequence is decreasing, and since the sequence is bounded below. If a sequence is decreasing and bounded below, then it has a limit. Then
has a limit and sequence (1), because 
11. if there are limits to functions 
.
L. Euler called this limit 
.
9. one-sided limits, function break.
number A is the left limit if the following holds for any sequence: .
number A is the right limit if the following holds for any sequence: .
If at the point A belonging to the domain of definition of the function or its boundary, the condition of continuity of the function is violated, then the point A is called a discontinuity point or discontinuity of a function. if, as the point tends
12. the sum of the terms of an infinite decreasing geometric progression.
Geometric progression is a sequence in which the ratio between the subsequent and previous terms remains unchanged, this ratio is called the denominator of the progression. Sum of first n members of the geometric progression is expressed by the formula 
This formula is convenient to use for a decreasing geometric progression - a progression in which the absolute value of its denominator is less than zero. 
- first member; 
- progression denominator; 
- number of the taken member of the sequence. The sum of an infinite decreasing progression is the number to which the sum of the first terms of a decreasing progression indefinitely approaches when the number increases indefinitely. 
That. The sum of the terms of an infinitely decreasing geometric progression is equal to 
.
Very easy to remember.
Well, let’s not go far, let’s look at it right away inverse function. Which function is the inverse of the exponential function? Logarithm:
In our case, the base is the number:
Such a logarithm (that is, a logarithm with a base) is called “natural”, and we use a special notation for it: we write instead.
What is it equal to? Of course, .
The derivative of the natural logarithm is also very simple:
Examples:
- Find the derivative of the function.
- What is the derivative of the function?
Answers: The exponential and natural logarithm are uniquely simple functions from a derivative perspective. Exponential and logarithmic functions with any other base will have a different derivative, which we will analyze later, after we go through the rules of differentiation.
Rules of differentiation
Rules of what? Again a new term, again?!...
Differentiation is the process of finding the derivative.
That's all. What else can you call this process in one word? Not derivative... The differential of mathematicians is the same increment of a function at. This term comes from the Latin differentia - difference. Here.
When deriving all these rules, we will use two functions, for example, and. We will also need formulas for their increments:
There are 5 rules in total.
The constant is taken out of the derivative sign.
If - some constant number (constant), then.
Obviously, this rule also works for the difference: .
Let's prove it. Let it be, or simpler.
Examples.
Find the derivatives of the functions:
- at a point;
- at a point;
- at a point;
- at the point.
Solutions:
- (the derivative is the same at all points, since it is a linear function, remember?);
Derivative of the product
Everything is similar here: let’s introduce a new function and find its increment:
Derivative:
Examples:
- Find the derivatives of the functions and;
- Find the derivative of the function at a point.
Solutions:
Derivative of an exponential function
Now your knowledge is enough to learn how to find the derivative of any exponential function, and not just exponents (have you forgotten what that is yet?).
So, where is some number.
We already know the derivative of the function, so let's try to reduce our function to a new base:
For this we will use simple rule: . Then:
Well, it worked. Now try to find the derivative, and don't forget that this function is complex.
Happened?
Here, check yourself:
The formula turned out to be very similar to the derivative of an exponent: as it was, it remains the same, only a factor appeared, which is just a number, but not a variable.
Examples:
Find the derivatives of the functions:
Answers:
This is just a number that cannot be calculated without a calculator, that is, it cannot be written down in a simpler form. Therefore, we leave it in this form in the answer.
Note that here is the quotient of two functions, so we apply the corresponding differentiation rule:
In this example, the product of two functions:
Derivative of a logarithmic function
It’s similar here: you already know the derivative of the natural logarithm:
Therefore, to find an arbitrary logarithm with a different base, for example:
We need to reduce this logarithm to the base. How do you change the base of a logarithm? I hope you remember this formula:
Only now we will write instead:
The denominator is simply a constant (a constant number, without a variable). The derivative is obtained very simply:
Derivatives of exponential and logarithmic functions are almost never found in the Unified State Examination, but it will not be superfluous to know them.
Derivative of a complex function.
What is a "complex function"? No, this is not a logarithm, and not an arctangent. These functions can be difficult to understand (although if you find the logarithm difficult, read the topic “Logarithms” and you will be fine), but from a mathematical point of view, the word “complex” does not mean “difficult”.
Imagine a small conveyor belt: two people are sitting and doing some actions with some objects. For example, the first one wraps a chocolate bar in a wrapper, and the second one ties it with a ribbon. The result is a composite object: a chocolate bar wrapped and tied with a ribbon. To eat a chocolate bar, you need to do the reverse steps in reverse order.
Let's create a similar mathematical pipeline: first we will find the cosine of a number, and then square the resulting number. So, we are given a number (chocolate), I find its cosine (wrapper), and then you square what I got (tie it with a ribbon). What happened? Function. This is an example of a complex function: when, to find its value, we perform the first action directly with the variable, and then a second action with what resulted from the first.
In other words, a complex function is a function whose argument is another function: .
For our example, .
We can easily do the same steps in reverse order: first you square it, and I then look for the cosine of the resulting number: . It’s easy to guess that the result will almost always be different. Important Feature complex functions: when the order of actions changes, the function changes.
Second example: (same thing). .
The action we do last will be called "external" function, and the action performed first - accordingly "internal" function(these are informal names, I use them only to explain the material in simple language).
Try to determine for yourself which function is external and which internal:
Answers: Separating inner and outer functions is very similar to changing variables: for example, in a function
- What action will we perform first? First, let's calculate the sine, and only then cube it. This means that it is an internal function, but an external one.
And the original function is their composition: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: . - Internal: ; external: .
Examination: .
We change variables and get a function.
Well, now we will extract our chocolate bar and look for the derivative. The procedure is always reversed: first we look for the derivative of the outer function, then we multiply the result by the derivative of the inner function. In relation to the original example, it looks like this:
Another example:
So, let's finally formulate the official rule:
Algorithm for finding the derivative of a complex function:
It seems simple, right?
Let's check with examples:
Solutions:
1) Internal: ;
External: ;
2) Internal: ;
(just don’t try to cut it by now! Nothing comes out from under the cosine, remember?)
3) Internal: ;
External: ;
It is immediately clear that this is a three-level complex function: after all, this is already a complex function in itself, and we also extract the root from it, that is, we perform the third action (put the chocolate in a wrapper and with a ribbon in the briefcase). But there is no reason to be afraid: we will still “unpack” this function in the same order as usual: from the end.
That is, first we differentiate the root, then the cosine, and only then the expression in brackets. And then we multiply it all.
In such cases, it is convenient to number the actions. That is, let's imagine what we know. In what order will we perform actions to calculate the value of this expression? Let's look at an example:
The later the action is performed, the more “external” the corresponding function will be. The sequence of actions is the same as before:
Here the nesting is generally 4-level. Let's determine the course of action.
1. Radical expression. .
2. Root. .
3. Sine. .
4. Square. .
5. Putting it all together:
DERIVATIVE. BRIEFLY ABOUT THE MAIN THINGS
Derivative of a function- the ratio of the increment of the function to the increment of the argument for an infinitesimal increment of the argument:
Basic derivatives:
Rules of differentiation:
The constant is taken out of the derivative sign:
Derivative of the sum:
Derivative of the product:
Derivative of the quotient:
Derivative of a complex function:
Algorithm for finding the derivative of a complex function:
- We define the “internal” function and find its derivative.
- We define the “external” function and find its derivative.
- We multiply the results of the first and second points.
Definition 1
If for each pair $(x,y)$ of values of two independent variables from some domain a certain value $z$ is associated, then $z$ is said to be a function of two variables $(x,y)$. Notation: $z=f(x,y)$.
In relation to the function $z=f(x,y)$, let's consider the concepts of general (total) and partial increments of a function.
Let a function $z=f(x,y)$ be given of two independent variables $(x,y)$.
Note 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 of $\Delta x$, 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 will give the variable $y$ an increment of $\Delta y$, 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 given an increment $\Delta x$, and the argument $y$ is given an increment $\Delta y$, then the full increment of the given function $z=f(x,y)$ is obtained. Designation:
Thus we have:
$\Delta _(x) z=f(x+\Delta x,y)-f(x,y)$ - partial increment of the function $z=f(x,y)$ by $x$;
$\Delta _(y) z=f(x,y+\Delta y)-f(x,y)$ - partial increment of the function $z=f(x,y)$ by $y$;
$\Delta z=f(x+\Delta x,y+\Delta y)-f(x,y)$ - total increment of the function $z=f(x,y)$.
Example 1
Solution:
$\Delta _(x) z=x+\Delta x+y$ - partial increment of the function $z=f(x,y)$ over $x$;
$\Delta _(y) z=x+y+\Delta y$ - 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 increment of the function $z=xy$ at the point $(1;2)$ for $\Delta x=0.1;\, \, \Delta y=0.1$.
Solution:
By definition of partial increment we find:
$\Delta _(x) z=(x+\Delta x)\cdot y$ - partial increment of the function $z=f(x,y)$ over $x$
$\Delta _(y) z=x\cdot (y+\Delta y)$ - partial increment of the function $z=f(x,y)$ by $y$;
By definition of total increment we find:
$\Delta z=(x+\Delta x)\cdot (y+\Delta y)$ - total increment of the function $z=f(x,y)$.
Hence,
\[\Delta _(x) z=(1+0.1)\cdot 2=2.2\] \[\Delta _(y) z=1\cdot (2+0.1)=2.1 \] \[\Delta z=(1+0.1)\cdot (2+0.1)=1.1\cdot 2.1=2.31.\]
Note 2
The total increment of a 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 assertion remarks for 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)
Let's find the sum of partial increments of a 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 $w$ is associated, then $w$ is said to be a function of three variables $(x,y,z)$ in this area.
Notation: $w=f(x,y,z)$.
Definition 3
If for each set $(x,y,z,...,t)$ of values of independent variables from some domain a certain value $w$ is associated, then $w$ is said to be a function of the variables $(x,y, z,...,t)$ in this area.
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 $w=f(x,y,z,...,t )$ by $z$;
$\Delta _(t) w=f(x,y,z,...,t+\Delta t)-f(x,y,z,...,t)$ - partial increment of the function $w=f (x,y,z,...,t)$ by $t$.
Example 4
Write partial and total increment functions
Solution:
By definition of partial increment we find:
$\Delta _(x) w=((x+\Delta x)+y)\cdot z$ - partial increment of the function $w=f(x,y,z)$ over $x$
$\Delta _(y) w=(x+(y+\Delta y))\cdot z$ - partial increment of the function $w=f(x,y,z)$ over $y$;
$\Delta _(z) w=(x+y)\cdot (z+\Delta z)$ - partial increment of the function $w=f(x,y,z)$ over $z$;
By definition of 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 increment 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 partial increment we find:
$\Delta _(x) w=(x+\Delta x)\cdot y\cdot z$ - partial increment of the function $w=f(x,y,z)$ over $x$
$\Delta _(y) w=x\cdot (y+\Delta y)\cdot z$ - partial increment of the function $w=f(x,y,z)$ by $y$;
$\Delta _(z) w=x\cdot y\cdot (z+\Delta z)$ - partial increment of the function $w=f(x,y,z)$ over $z$;
By definition of 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,
\[\Delta _(x) w=(1+0.1)\cdot 2\cdot 1=2.2\] \[\Delta _(y) w=1\cdot (2+0.1)\ cdot 1=2.1\] \[\Delta _(y) w=1\cdot 2\cdot (1+0.1)=2.2\] \[\Delta z=(1+0.1) \cdot (2+0.1)\cdot (1+0.1)=1.1\cdot 2.1\cdot 1.1=2.541.\]
From a geometric 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 function $z=f(x,y)$ when moving from point $M(x,y)$ to point $M_(1) (x+\Delta x,y+\Delta y)$ (Fig. 1).
Picture 1.
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