The derivative of the function f (x) at the point x0 is the limit (if it exists) of the ratio of the increment of the function at the point x0 to the increment of the argument Δx, if the increment of the argument tends to zero and is denoted by f ‘(x0). The action of finding the derivative of a function is called differentiation.
The derivative of a function has the following physical meaning: the derivative of a function at a given point is the rate of change of the function at a given point.
The geometric meaning of the derivative. The derivative at the point x0 is equal to the slope of the tangent to the graph of the function y=f(x) at this point.
The physical meaning of the derivative. If a point moves along the x-axis and its coordinate changes according to the x(t) law, then the instantaneous speed of the point:
The concept of a differential, its properties. Differentiation rules. Examples.
Definition. The differential of a function at some point x is the main, linear part of the increment of the function. The differential of the function y = f(x) is equal to the product of its derivative and the increment of the independent variable x (argument).
It is written like this:
or 
Or 
Differential Properties
The differential has properties similar to those of the derivative: 
To basic rules of differentiation include:
1) taking the constant factor out of the sign of the derivative
2) derivative of the sum, derivative of the difference
3) derivative of the product of functions
4) derivative of a quotient of two functions (derivative of a fraction) 
Examples.
Let's prove the formula: By the definition of the derivative, we have: 
An arbitrary factor can be taken out of the sign of the passage to the limit (this is known from the properties of the limit), therefore
For example: Find the derivative of a function
Decision: We use the rule of taking the multiplier out of the sign of the derivative :
Quite often, it is necessary to first simplify the form of a differentiable function in order to use the table of derivatives and the rules for finding derivatives. The following examples clearly confirm this.
Differentiation formulas. Application of the differential in approximate calculations. Examples.
The use of the differential in approximate calculations allows the use of the differential for approximate calculations of function values.
Examples.
Using the differential, calculate approximately
To calculate this value, we apply the formula from the theory
Let us introduce a function and represent the given value in the form
then Calculate 
Substituting everything into the formula, we finally get
Answer: 
16. L'Hopital's rule for disclosure of uncertainties of the form 0/0 Or ∞/∞. Examples.
The limit of the ratio of two infinitesimal or two infinitely large quantities is equal to the limit of the ratio of their derivatives. 
1) 
17. Increasing and decreasing functions. extremum of the function. Algorithm for studying a function for monotonicity and extremum. Examples.
Function increases on an interval if for any two points of this interval related by the relation , the inequality is true. That is, a larger value of the argument corresponds to a larger value of the function, and its graph goes “from bottom to top”. The demo function grows over the interval
Likewise, the function decreases on an interval if for any two points of the given interval, such that , the inequality is true. That is, a larger value of the argument corresponds to a smaller value of the function, and its graph goes “from top to bottom”. Ours decreases on intervals decreases on intervals 
.
Extremes The point is called the maximum point of the function y=f(x) if the inequality is true for all x from its neighborhood. The value of the function at the maximum point is called function maximum and denote .
The point is called the minimum point of the function y=f(x) if the inequality is true for all x from its neighborhood. The value of the function at the minimum point is called function minimum and denote .
The neighborhood of a point is understood as the interval 
, where is a sufficiently small positive number.
The minimum and maximum points are called extremum points, and the function values corresponding to the extremum points are called function extrema.
To explore a function for monotony use the following diagram:
- Find the scope of the function;
- Find the derivative of the function and the domain of the derivative;
- Find the zeros of the derivative, i.e. the value of the argument at which the derivative is equal to zero;
- On the numerical beam, mark the common part of the domain of the function and the domain of its derivative, and on it - the zeros of the derivative;
- Determine the signs of the derivative on each of the obtained intervals;
- By the signs of the derivative, determine at which intervals the function increases and at which it decreases;
- Record the appropriate gaps separated by semicolons.
Algorithm for studying a continuous function y = f(x) for monotonicity and extrema:
1) Find the derivative f ′(x).
2) Find stationary (f ′(x) = 0) and critical (f ′(x) does not exist) points of the function y = f(x).
3) Mark the stationary and critical points on the number line and determine the signs of the derivative on the resulting intervals.
4) Draw conclusions about the monotonicity of the function and its extremum points.
18. Convexity of a function. Inflection points. Algorithm for examining a function for convexity (Concavity) Examples.
convex down on the X interval, if its graph is located not lower than the tangent to it at any point of the X interval.
The differentiable function is called convex up on the X interval, if its graph is located no higher than the tangent to it at any point of the X interval.
The point formula is called graph inflection point function y \u003d f (x), if at a given point there is a tangent to the graph of the function (it can be parallel to the Oy axis) and there is such a neighborhood of the point formula, within which the graph of the function has different directions of convexity to the left and to the right of the point M.
Finding intervals for convexity:
If the function y=f(x) has a finite second derivative on the interval X and if the inequality 
(), then the graph of the function has a convexity directed down (up) on X.
This theorem allows you to find the intervals of concavity and convexity of a function, you only need to solve the inequalities and, respectively, on the domain of definition of the original function.
Example: Find out the intervals at which the graph of the functionFind out the intervals at which the graph of the function 
has a convexity directed upwards and a convexity directed downwards. has a convexity directed upwards and a convexity directed downwards.
Decision: The domain of this function is the entire set of real numbers.
Let's find the second derivative.
The domain of definition of the second derivative coincides with the domain of definition of the original function, therefore, in order to find out the intervals of concavity and convexity, it is enough to solve and respectively. 
Therefore, the function is downward convex on the interval formula and upward convex on the interval formula.
19) Asymptotes of a function. Examples.
Direct called vertical asymptote graph of the function if at least one of the limit values or is equal to or .
Comment. The line cannot be a vertical asymptote if the function is continuous at . Therefore, vertical asymptotes should be sought at the discontinuity points of the function.
Direct called horizontal asymptote graph of the function if at least one of the limit values or is equal to .
Comment. A function graph can only have a right horizontal asymptote or only a left one.
Direct called oblique asymptote graph of the function if
EXAMPLE:
Exercise. Find asymptotes of the graph of a function 
Decision. Function scope:
a) vertical asymptotes: a straight line is a vertical asymptote, since
b) horizontal asymptotes: we find the limit of the function at infinity:
that is, there are no horizontal asymptotes.
c) oblique asymptotes:
Thus, the oblique asymptote is: .
Answer. The vertical asymptote is a straight line.
The oblique asymptote is a straight line.
20) The general scheme of the study of the function and plotting. Example.
a.
Find the ODZ and breakpoints of the function.
b. Find the points of intersection of the graph of the function with the coordinate axes.
2. Conduct a study of the function using the first derivative, that is, find the extremum points of the function and the intervals of increase and decrease.
3. Investigate the function using the second-order derivative, that is, find the inflection points of the function graph and the intervals of its convexity and concavity.
4. Find the asymptotes of the graph of the function: a) vertical, b) oblique.
5. On the basis of the study, build a graph of the function.
Note that before plotting, it is useful to establish whether a given function is even or odd.
Recall that a function is called even if the value of the function does not change when the sign of the argument changes: f(-x) = f(x) and a function is called odd if f(-x) = -f(x).
In this case, it suffices to study the function and build its graph for positive values of the argument belonging to the ODZ. With negative values of the argument, the graph is completed on the basis that for an even function it is symmetrical about the axis Oy, and for odd with respect to the origin.
Examples. Explore functions and build their graphs.
Function scope D(y)= (–∞; +∞). There are no break points.
Axis intersection Ox: x = 0,y= 0.
The function is odd, therefore, it can be investigated only on an interval equal to the ratio of the distance traveled during this period of time to the time, i.e.
Vav = ∆x /∆t . Let us pass to the limit in the last equality as ∆ t → 0.
lim V cf (t) = n (t 0 ) - instantaneous speed at time t 0 , ∆t → 0.
and lim \u003d ∆ x / ∆ t \u003d x "(t 0 ) (by definition of a derivative).
So, n(t) = x "(t).
The physical meaning of the derivative is as follows: the derivative of the function y = f( x) at the pointx 0 is the rate of change of the function f(x) at the pointx 0
The derivative is used in physics to find the speed from a known function of coordinates from time, acceleration from a known function of speed from time.
u (t) \u003d x "(t) - speed,
a(f) = n "(t ) - acceleration, or
a (t) \u003d x "(t).
If the law of motion of a material point along a circle is known, then it is possible to find the angular velocity and angular acceleration during rotational motion:
φ = φ (t ) - change of angle from time,
ω = φ "(t ) - angular velocity,
ε = φ "(t ) - angular acceleration, orε \u003d φ "(t).
If the distribution law for the mass of an inhomogeneous rod is known, then the linear density of the inhomogeneous rod can be found:
m \u003d m (x) - mass,
x н , l - rod length,
p = m "(x) - linear density.
With the help of the derivative, problems from the theory of elasticity and harmonic vibrations are solved. Yes, according to Hooke's law
F = - kx , x - variable coordinate, k - coefficient of elasticity of the spring. Puttingω 2 = k / m , we obtain the differential equation of the spring pendulum x "( t ) + ω 2 x(t ) = 0,
where ω = √k /√m oscillation frequency ( l/c ), k - spring stiffness ( H/m).
An equation of the form y" +ω 2 y = 0 is called the equation of harmonic oscillations (mechanical, electrical, electromagnetic). The solution of such equations is the function
y \u003d Asin (ωt + φ 0 ) or y \u003d Acos (ωt + φ 0 ), where
A is the amplitude of oscillations,ω - cyclic frequency,
φ 0 - initial phase.
Subject. Derivative. Geometric and mechanical meaning of the derivative
If this limit exists, then the function is said to be differentiable at a point. The derivative of a function is denoted (formula 2).
- The geometric meaning of the derivative. Consider the function graph. It can be seen from Fig. 1 that for any two points A and B of the graph of the function, formula 3) can be written. In it - the angle of inclination of the secant AB.
Thus, the difference ratio is equal to the slope of the secant. If we fix point A and move point B towards it, then it decreases indefinitely and approaches 0, and the secant AB approaches the tangent AC. Therefore, the limit of the difference ratio is equal to the slope of the tangent at point A. Hence the conclusion follows.
The derivative of a function at a point is the slope of the tangent to the graph of that function at that point. This is the geometric meaning of the derivative.
- Tangent equation . Let's derive the equation of the tangent to the graph of the function at the point. In the general case, the equation of a straight line with a slope has the form: . To find b, we use the fact that the tangent passes through the point A: . This implies: . Substituting this expression for b, we obtain the tangent equation (formula 4).
Consider the graph of some function y = f(x).
We mark on it some point A with coordinates (x, f (x)) and not far from it a point B with coordinates (x + h, f (x + h). Draw a line (AB) through these points. Consider the expression 
. The difference f(x+h)-f(x) is equal to the distance BL, and the distance AL is equal to h. The ratio BL/AL is the tangent ε of the angle - the angle of inclination of the straight line (AB). Now imagine that h is very, very small. Then the line (AB) will almost coincide with the tangent at the point x to the graph of the function y = f(x).
So, let's give definitions.
The derivative of the function y = f(x) at the point x is called the limit of the relation as h tends to zero. Write: 
The geometric meaning of the derivative is the tangent of the slope of the tangent.
The derivative also has a physical meaning. In elementary grades, speed was defined as distance divided by time. However, in real life, the speed of, for example, a car is not constant throughout the entire journey. Let the path be some function of time - S(t). Let's fix the moment of time t. In a short period of time from t to t + h, the car will cover the path S(t+h)-S(t). For a short period of time, the speed will not change much and therefore, you can use the definition of speed known from elementary school 
. And as h tends to zero, this will be the derivative.