Solving the derivative for dummies: definition, how to find, examples of solutions. Function derivative
(\large\bf Function derivative)
Consider the function y=f(x), given on the interval (a,b). Let x- any fixed point interval (a,b), a Δx- an arbitrary number, such that the value x+Δx also belongs to the interval (a,b). This number Δx is called argument increment.
Definition. Function increment y=f(x) at the point x, corresponding to the increment of the argument Δx, let's call the number
Δy = f(x+Δx) - f(x).
We believe that Δx ≠ 0. Consider at a given fixed point x the ratio of the increment of the function at that point to the corresponding increment of the argument Δx
This relation will be called the difference relation. Since the value x we consider fixed, the difference relation is a function of the argument Δx. This function is defined for all argument values Δx, belonging to some sufficiently small neighborhood of the point ∆x=0, except for the point ∆x=0. Thus, we have the right to consider the question of the existence of a limit of the specified function for ∆x → 0.
Definition. Derivative function y=f(x) at a given fixed point x is called the limit ∆x → 0 differential relation, that is
Provided that this limit exists.
Designation. y (x) or f′(x).
The geometric meaning of the derivative: Derivative of function f(x) at this point x equal to the tangent of the angle between the axis Ox and a tangent to the graph of this function at the corresponding point:
f′(x 0) = \tgα.
The mechanical meaning of the derivative: The derivative of the path with respect to time is equal to the speed of the rectilinear movement of the point:
Line tangent equation y=f(x) at the point M0 (x0,y0) takes the form
y-y 0 = f (x 0) (x-x 0).
The normal to the curve at some point is the perpendicular to the tangent at the same point. If a f′(x 0)≠ 0, then the equation of the normal to the line y=f(x) at the point M0 (x0,y0) is written like this:
The concept of differentiability of a function
Let the function y=f(x) defined on some interval (a,b), x- some fixed value of the argument from this interval, Δx- any increment of the argument such that the value of the argument x+Δx ∈ (a, b).
Definition. Function y=f(x) is called differentiable at a given point x if increment Δy this function at the point x, corresponding to the increment of the argument Δx, can be represented as
Δy = A Δx +αΔx,
where A is some number independent of Δx, a α - argument function Δx, which is infinitely small at ∆x → 0.
Since the product of two infinitesimal functions αΔx is an infinitesimal higher order than Δx(property 3 of infinitesimal functions), we can write:
∆y = A ∆x +o(∆x).
Theorem. In order for the function y=f(x) was differentiable at a given point x, it is necessary and sufficient that it has a finite derivative at this point. Wherein A=f′(x), that is
Δy = f′(x) Δx +o(Δx).
The operation of finding the derivative is usually called differentiation.
Theorem. If the function y=f(x) x, then it is continuous at that point.
Comment. From the continuity of the function y=f(x) at this point x, generally speaking, it does not follow that the function is differentiable f(x) at this point. For example, the function y=|x|- continuous at a point x=0, but has no derivative.
The concept of a function differential
Definition. function differential y=f(x) is called the product of the derivative of this function and the increment of the independent variable x:
dy = y′ ∆x, df(x) = f′(x) ∆x.
For function y=x we get dy=dx=x'Δx = 1 Δx= Δx, that is dx=Δx- the differential of an independent variable is equal to the increment of this variable.
Thus, we can write
dy = y′dx, df(x) = f′(x)dx
Differential dy and increment Δy functions y=f(x) at this point x, both corresponding to the same increment of the argument Δx are, in general, not equal to each other.
The geometric meaning of the differential: The differential of a function is equal to the increment of the ordinate of the tangent to the graph of the given function when the argument is incremented Δx.
Differentiation rules
Theorem. If each of the functions u(x) and v(x) differentiable at a given point x, then the sum, difference, product and quotient of these functions (quotient provided that v(x)≠ 0) are also differentiable at this point, and the following formulas hold:
Consider a complex function y=f(φ(x))≡ F(x), where y=f(u), u=φ(x). In this case u called intermediate argument, x - independent variable.
Theorem. If a y=f(u) and u=φ(x) are differentiable functions of their arguments, then the derivative of the complex function y=f(φ(x)) exists and is equal to the product of this function with respect to the intermediate argument and the derivative of the intermediate argument with respect to the independent variable, i.e.
Comment. For a complex function that is a superposition of three functions y=F(f(φ(x))), the differentiation rule has the form
y′ x = y′ u u′ v v′ x,
where functions v=φ(x), u=f(v) and y=F(u) are differentiable functions of their arguments.
Theorem. Let the function y=f(x) is increasing (or decreasing) and continuous in some neighborhood of the point x0. Let, in addition, this function be differentiable at the indicated point x0 and its derivative at this point f′(x 0) ≠ 0. Then in some neighborhood of the corresponding point y0=f(x0) the inverse for y=f(x) function x=f -1 (y), and the indicated inverse function is differentiable at the corresponding point y0=f(x0) and for its derivative at this point y the formula is valid
Derivative table
Invariance of the form of the first differential
Consider the differential of a complex function. If a y=f(x), x=φ(t) are differentiable functions of their arguments, then the derivative of the function y=f(φ(t)) is expressed by the formula
y′ t = y′ x x′ t.
By definition dy=y't dt, then we get
dy = y′ t dt = y′ x x′ t dt = y′ x (x′ t dt) = y′ x dx,
dy = y′ x dx.
So, we have proven
Property of invariance of the form of the first differential of a function: as in the case when the argument x is an independent variable, and in the case when the argument x is itself a differentiable function of the new variable, the differential dy functions y=f(x) is equal to the derivative of this function, multiplied by the differential of the argument dx.
Application of the differential in approximate calculations
We have shown that the differential dy functions y=f(x), generally speaking, is not equal to the increment Δy this function. Nevertheless, up to an infinitely small function of a higher order of smallness than Δx, the approximate equality
∆y ≈ dy.
The ratio is called the relative error of the equality of this equality. Because ∆y-dy=o(∆x), then the relative error of this equality becomes arbitrarily small as |Δх|.
Given that Δy=f(x+δx)-f(x), dy=f′(x)Δx, we get f(x+δx)-f(x) ≈ f′(x)Δx or
f(x+δx) ≈ f(x) + f′(x)Δx.
This approximate equality allows with an error o(Δx) replace function f(x) in a small neighborhood of a point x(i.e. for small values Δx) a linear function of the argument Δx standing on the right side.
Derivatives of higher orders
Definition. The second derivative (or second order derivative) of the function y=f(x) is called the derivative of its first derivative.
Notation for the second derivative of a function y=f(x):
Mechanical meaning of the second derivative. If the function y=f(x) describes the law of motion of a material point in a straight line, then the second derivative f″(x) is equal to the acceleration of the moving point at time x.
The third and fourth derivatives are defined similarly.
Definition. n-th derivative (or derivative n th order) functions y=f(x) called the derivative of it n-1-th derivative:
y (n) =(y (n-1))′, f (n) (x)=(f (n-1) (x))′.
Designations: y″′, y IV, y V etc.
It is absolutely impossible to solve physical problems or examples in mathematics without knowledge about the derivative and methods for calculating it. The derivative is one of the most important concepts of mathematical analysis. We decided to devote today's article to this fundamental topic. What is a derivative, what is its physical and geometric meaning, how to calculate the derivative of a function? All these questions can be combined into one: how to understand the derivative?
Geometric and physical meaning of the derivative
Let there be a function f(x) , given in some interval (a,b) . The points x and x0 belong to this interval. When x changes, the function itself changes. Argument change - difference of its values x-x0 . This difference is written as delta x and is called argument increment. The change or increment of a function is the difference between the values of the function at two points. Derivative definition:
The derivative of a function at a point is the limit of the ratio of the increment of the function at a given point to the increment of the argument when the latter tends to zero.
Otherwise it can be written like this:
What is the point in finding such a limit? But which one:
the derivative of a function at a point is equal to the tangent of the angle between the OX axis and the tangent to the graph of the function at a given point.
The physical meaning of the derivative: the time derivative of the path is equal to the speed of the rectilinear motion.
Indeed, since school days, everyone knows that speed is a private path. x=f(t) and time t . Average speed over a certain period of time:
To find out the speed of movement at a time t0 you need to calculate the limit:
Rule one: take out the constant
The constant can be taken out of the sign of the derivative. Moreover, it must be done. When solving examples in mathematics, take as a rule - if you can simplify the expression, be sure to simplify .
Example. Let's calculate the derivative:
Rule two: derivative of the sum of functions
The derivative of the sum of two functions is equal to the sum of the derivatives of these functions. The same is true for the derivative of the difference of functions.
We will not give a proof of this theorem, but rather consider a practical example.
Find the derivative of a function:
Rule three: the derivative of the product of functions
The derivative of the product of two differentiable functions is calculated by the formula:
Example: find the derivative of a function:
Solution: 
Here it is important to say about the calculation of derivatives of complex functions. The derivative of a complex function is equal to the product of the derivative of this function with respect to the intermediate argument by the derivative of the intermediate argument with respect to the independent variable.
In the above example, we encounter the expression:
In this case, the intermediate argument is 8x to the fifth power. In order to calculate the derivative of such an expression, we first consider the derivative of the external function with respect to the intermediate argument, and then multiply by the derivative of the intermediate argument itself with respect to the independent variable.
Rule Four: The derivative of the quotient of two functions
Formula for determining the derivative of a quotient of two functions:
We tried to talk about derivatives for dummies from scratch. This topic is not as simple as it seems, so be warned: there are often pitfalls in the examples, so be careful when calculating derivatives.
With any question on this and other topics, you can contact the student service. In a short time, we will help you solve the most difficult control and deal with tasks, even if you have never dealt with the calculation of derivatives before.
The operation of finding a derivative is called differentiation.
As a result of solving problems of finding derivatives of the simplest (and not very simple) functions by defining the derivative as the limit of the ratio of the increment to the increment of the argument, a table of derivatives and precisely defined rules of differentiation appeared. Isaac Newton (1643-1727) and Gottfried Wilhelm Leibniz (1646-1716) were the first to work in the field of finding derivatives.
Therefore, in our time, in order to find the derivative of any function, it is not necessary to calculate the above-mentioned limit of the ratio of the increment of the function to the increment of the argument, but only need to use the table of derivatives and the rules of differentiation. The following algorithm is suitable for finding the derivative.
To find the derivative, you need an expression under the stroke sign break down simple functions and determine what actions (product, sum, quotient) these functions are related. Further, we find the derivatives of elementary functions in the table of derivatives, and the formulas for the derivatives of the product, sum and quotient - in the rules of differentiation. The table of derivatives and differentiation rules are given after the first two examples.
Example 1 Find the derivative of a function
Solution. From the rules of differentiation we find out that the derivative of the sum of functions is the sum of derivatives of functions, i.e.
From the table of derivatives, we find out that the derivative of "X" is equal to one, and the derivative of the sine is cosine. We substitute these values in the sum of derivatives and find the derivative required by the condition of the problem:
Example 2 Find the derivative of a function
Solution. Differentiate as a derivative of the sum, in which the second term with a constant factor, it can be taken out of the sign of the derivative:
If there are still questions about where something comes from, they, as a rule, become clear after reading the table of derivatives and the simplest rules of differentiation. We are going to them right now.
Table of derivatives of simple functions
| 1. Derivative of a constant (number). Any number (1, 2, 5, 200...) that is in the function expression. Always zero. This is very important to remember, as it is required very often | |
| 2. Derivative of the independent variable. Most often "x". Always equal to one. This is also important to remember | |
| 3. Derivative of degree. When solving problems, you need to convert non-square roots to a power. | |
| 4. Derivative of a variable to the power of -1 | |
| 5. Derivative of the square root | |
| 6. Sine derivative | |
| 7. Cosine derivative |  |
| 8. Tangent derivative |  |
| 9. Derivative of cotangent |  |
| 10. Derivative of the arcsine |  |
| 11. Derivative of arc cosine |  |
| 12. Derivative of arc tangent |  |
| 13. Derivative of the inverse tangent |  |
| 14. Derivative of natural logarithm | |
| 15. Derivative of a logarithmic function |  |
| 16. Derivative of the exponent | |
| 17. Derivative of exponential function |
Differentiation rules
| 1. Derivative of the sum or difference |  |
| 2. Derivative of a product |  |
| 2a. Derivative of an expression multiplied by a constant factor | |
| 3. Derivative of the quotient |  |
| 4. Derivative of a complex function |  |
Rule 1If functions
are differentiable at some point , then at the same point the functions
and
those. the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions.
Consequence. If two differentiable functions differ by a constant, then their derivatives are, i.e.
Rule 2If functions
are differentiable at some point , then their product is also differentiable at the same point
and
those. the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other.
Consequence 1. The constant factor can be taken out of the sign of the derivative:
Consequence 2. The derivative of the product of several differentiable functions is equal to the sum of the products of the derivative of each of the factors and all the others.
For example, for three multipliers:
Rule 3If functions
differentiable at some point and , then at this point their quotient is also differentiable.u/v , and
those. the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator.
Where to look on other pages
When finding the derivative of the product and the quotient in real problems, it is always necessary to apply several differentiation rules at once, so more examples on these derivatives are in the article."The derivative of a product and a quotient".
Comment. You should not confuse a constant (that is, a number) as a term in the sum and as a constant factor! In the case of a term, its derivative is equal to zero, and in the case of a constant factor, it is taken out of the sign of the derivatives. This is a typical mistake that occurs at the initial stage of studying derivatives, but as the average student solves several one-two-component examples, the average student no longer makes this mistake.
And if, when differentiating a product or a quotient, you have a term u"v, wherein u- a number, for example, 2 or 5, that is, a constant, then the derivative of this number will be equal to zero and, therefore, the entire term will be equal to zero (such a case is analyzed in example 10).
Another common mistake is the mechanical solution of the derivative of a complex function as the derivative of a simple function. That's why derivative of a complex function devoted to a separate article. But first we will learn to find derivatives of simple functions.
Along the way, you can not do without transformations of expressions. To do this, you may need to open in new windows manuals Actions with powers and roots and Actions with fractions .
If you are looking for solutions to derivatives with powers and roots, that is, when the function looks like 
, then follow the lesson " Derivative of the sum of fractions with powers and roots".
If you have a task like 
, then you are in the lesson "Derivatives of simple trigonometric functions".
Step by step examples - how to find the derivative
Example 3 Find the derivative of a function
Solution. We determine the parts of the function expression: the entire expression represents the product, and its factors are sums, in the second of which one of the terms contains a constant factor. We apply the product differentiation rule: the derivative of the product of two functions is equal to the sum of the products of each of these functions and the derivative of the other:
Next, we apply the rule of differentiation of the sum: the derivative of the algebraic sum of functions is equal to the algebraic sum of the derivatives of these functions. In our case, in each sum, the second term with a minus sign. In each sum, we see both an independent variable, the derivative of which is equal to one, and a constant (number), the derivative of which is equal to zero. So, "x" turns into one, and minus 5 - into zero. In the second expression, "x" is multiplied by 2, so we multiply two by the same unit as the derivative of "x". We get the following values of derivatives:
We substitute the found derivatives into the sum of products and obtain the derivative of the entire function required by the condition of the problem:
And you can check the solution of the problem on the derivative on .
Example 4 Find the derivative of a function
Solution. We are required to find the derivative of the quotient. We apply the formula for differentiating a quotient: the derivative of a quotient of two functions is equal to a fraction whose numerator is the difference between the products of the denominator and the derivative of the numerator and the numerator and the derivative of the denominator, and the denominator is the square of the former numerator. We get:
We have already found the derivative of the factors in the numerator in Example 2. Let's also not forget that the product, which is the second factor in the numerator in the current example, is taken with a minus sign:
If you are looking for solutions to such problems in which you need to find the derivative of a function, where there is a continuous pile of roots and degrees, such as, for example, 
then welcome to class "The derivative of the sum of fractions with powers and roots" .
If you need to learn more about the derivatives of sines, cosines, tangents and other trigonometric functions, that is, when the function looks like , then you have a lesson "Derivatives of simple trigonometric functions" .
Example 5 Find the derivative of a function
Solution. In this function, we see a product, one of the factors of which is the square root of the independent variable, with the derivative of which we familiarized ourselves in the table of derivatives. According to the product differentiation rule and the tabular value of the derivative of the square root, we get:
You can check the solution of the derivative problem on derivative calculator online .
Example 6 Find the derivative of a function
Solution. In this function, we see the quotient, the dividend of which is the square root of the independent variable. According to the rule of differentiation of the quotient, which we repeated and applied in example 4, and the tabular value of the derivative of the square root, we get:
To get rid of the fraction in the numerator, multiply the numerator and denominator by .
The process of finding the derivative of a function is called differentiation. The derivative has to be found in a number of problems in the course of mathematical analysis. For example, when finding extremum points and inflection points of a function graph.
How to find?
To find the derivative of a function, you need to know the table of derivatives of elementary functions and apply the basic rules of differentiation:
- Taking the constant out of the sign of the derivative: $$ (Cu)" = C(u)" $$
- Derivative of sum/difference of functions: $$ (u \pm v)" = (u)" \pm (v)" $$
- Derivative of the product of two functions: $$ (u \cdot v)" = u"v + uv" $$
- Fraction derivative : $$ \bigg (\frac(u)(v) \bigg)" = \frac(u"v - uv")(v^2) $$
- Compound function derivative : $$ (f(g(x)))" = f"(g(x)) \cdot g"(x) $$
Solution examples
| Example 1 |
| Find the derivative of the function $ y = x^3 - 2x^2 + 7x - 1 $ |
| Solution |
|
The derivative of the sum/difference of functions is equal to the sum/difference of the derivatives: $$ y" = (x^3 - 2x^2 + 7x - 1)" = (x^3)" - (2x^2)" + (7x)" - (1)" = $$ Using the power function derivative rule $ (x^p)" = px^(p-1) $ we have: $$ y" = 3x^(3-1) - 2 \cdot 2 x^(2-1) + 7 - 0 = 3x^2 - 4x + 7 $$ It was also taken into account that the derivative of the constant is equal to zero. If you cannot solve your problem, then send it to us. We will provide a detailed solution. You will be able to familiarize yourself with the progress of the calculation and gather information. This will help you get a credit from the teacher in a timely manner! |
| Answer |
| $$ y" = 3x^2 - 4x + 7 $$ |
Derivative
Calculating the derivative of a mathematical function (differentiation) is a very common task in solving higher mathematics. For simple (elementary) mathematical functions, this is a fairly simple matter, since tables of derivatives for elementary functions have long been compiled and are easily accessible. However, finding the derivative of a complex mathematical function is not a trivial task and often requires significant effort and time.
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