Derivative of a function. The Ultimate Guide (2019)
Let's imagine a straight road passing through a hilly area. That is, it goes up and down, but does not turn right or left. If the axis is directed horizontally along the road and vertically, then the road line will be very similar to the graph of some continuous function:
The axis is a certain level of zero altitude; in life we use sea level as it.
As we move forward along such a road, we also move up or down. We can also say: when the argument changes (movement along the abscissa axis), the value of the function changes (movement along the ordinate axis). Now let's think about how to determine the “steepness” of our road? What kind of value could this be? It’s very simple: how much the height will change when moving forward a certain distance. Indeed, on different sections of the road, moving forward (along the x-axis) by one kilometer, we will rise or fall by a different number of meters relative to sea level (along the y-axis).
Let’s denote progress (read “delta x”).
The Greek letter (delta) is commonly used in mathematics as a prefix meaning "change". That is - this is a change in quantity, - a change; then what is it? That's right, a change in magnitude.
Important: an expression is a single whole, one variable. Never separate the “delta” from the “x” or any other letter!
That is, for example, .
So, we have moved forward, horizontally, by. If we compare the line of the road with the graph of the function, then how do we denote the rise? Certainly, . That is, as we move forward, we rise higher.
Let's return to "steepness": this is a value that shows how much (steeply) the height increases when moving forward one unit of distance:
Let us assume that on some section of the road, when moving forward by a kilometer, the road rises up by a kilometer. Then the slope at this place is equal. And if the road, while moving forward by m, dropped by km? Then the slope is equal.
Now let's look at the top of a hill. If you take the beginning of the section half a kilometer before the summit, and the end half a kilometer after it, you can see that the height is almost the same.
That is, according to our logic, it turns out that the slope here is almost equal to zero, which is clearly not true. Just over a distance of kilometers a lot can change. It is necessary to consider smaller areas for a more adequate and accurate assessment of steepness. For example, if you measure the change in height as you move one meter, the result will be much more accurate. But even this accuracy may not be enough for us - after all, if there is a pole in the middle of the road, we can simply pass it. What distance should we choose then? Centimeter? Millimeter? Less is better!
IN real life Measuring distances to the nearest millimeter is more than enough. But mathematicians always strive for perfection. Therefore, the concept was invented infinitesimal, that is, the absolute value is less than any number that we can name. For example, you say: one trillionth! How much less? And you divide this number by - and it will be even less. And so on. If we want to write that a quantity is infinitesimal, we write like this: (we read “x tends to zero”). It is very important to understand that this number is not equal to zero! But very close to it. This means that you can divide by it.
The concept opposite to infinitesimal is infinitely large (). You've probably already come across it when you were working on inequalities: this number is modulo greater than any number you can think of. If you come up with the biggest number possible, just multiply it by two and you'll get an even bigger number. And infinity is even greater than what happens. In fact, the infinitely large and the infinitely small are the inverse of each other, that is, at, and vice versa: at.
Now let's get back to our road. The ideally calculated slope is the slope calculated for an infinitesimal segment of the path, that is:
I note that with an infinitesimal displacement, the change in height will also be infinitesimal. But let me remind you that infinitesimal does not mean equal to zero. If you divide infinitesimal numbers by each other, you can get a completely ordinary number, for example, . That is, one small value can be exactly times larger than another.
What is all this for? The road, the steepness... We’re not going on a car rally, but we’re teaching mathematics. And in mathematics everything is exactly the same, only called differently.
Concept of derivative
The derivative of a function is the ratio of the increment of the function to the increment of the argument for an infinitesimal increment of the argument.
Incrementally in mathematics they call change. The extent to which the argument () changes as it moves along the axis is called argument increment and is designated. How much the function (height) has changed when moving forward along the axis by a distance is called function increment and is designated.
So, the derivative of a function is the ratio to when. We denote the derivative with the same letter as the function, only with a prime on the top right: or simply. So, let's write the derivative formula using these notations:
As in the analogy with the road, here when the function increases, the derivative is positive, and when it decreases, it is negative.
Can the derivative be equal to zero? Certainly. For example, if we are driving on a flat horizontal road, the steepness is zero. And it’s true, the height doesn’t change at all. So it is with the derivative: the derivative of a constant function (constant) is equal to zero:
since the increment of such a function is equal to zero for any.
Let's remember the hilltop example. It turned out that it was possible to arrange the ends of the segment on opposite sides of the vertex in such a way that the height at the ends turns out to be the same, that is, the segment is parallel to the axis:
But large segments are a sign of inaccurate measurement. We will raise our segment up parallel to itself, then its length will decrease.
Eventually, when we are infinitely close to the top, the length of the segment will become infinitesimal. But at the same time, it remained parallel to the axis, that is, the difference in heights at its ends is equal to zero (it does not tend to, but is equal to). So the derivative
This can be understood this way: when we stand at the very top, a small shift to the left or right changes our height negligibly.
There is also a purely algebraic explanation: to the left of the vertex the function increases, and to the right it decreases. As we found out earlier, when a function increases, the derivative is positive, and when it decreases, it is negative. But it changes smoothly, without jumps (since the road does not change its slope sharply anywhere). Therefore, there must be between negative and positive values. It will be where the function neither increases nor decreases - at the vertex point.
The same is true for the trough (the area where the function on the left decreases and on the right increases):
A little more about increments.
So we change the argument to magnitude. We change from what value? What has it (the argument) become now? We can choose any point, and now we will dance from it.
Consider a point with a coordinate. The value of the function in it is equal. Then we do the same increment: we increase the coordinate by. What now? equal argument? Very easy: . What is the value of the function now? Where the argument goes, so does the function: . What about function increment? Nothing new: this is still the amount by which the function has changed:
Practice finding increments:
- Find the increment of the function at a point when the increment of the argument is equal to.
- The same goes for the function at a point.
Solutions:
At different points with the same argument increment, the function increment will be different. This means that the derivative at each point is different (we discussed this at the very beginning - the steepness of the road is different at different points). Therefore, when we write a derivative, we must indicate at what point:
Power function.
A power function is a function where the argument is to some degree (logical, right?).
Moreover - to any extent: .
The simplest case is when the exponent is:
Let's find its derivative at a point. Let's recall the definition of a derivative:
So the argument changes from to. What is the increment of the function?
Increment is this. But a function at any point is equal to its argument. That's why:
The derivative is equal to:
The derivative of is equal to:
b) Now consider quadratic function (): .
Now let's remember that. This means that the value of the increment can be neglected, since it is infinitesimal, and therefore insignificant against the background of the other term:
So, we came up with another rule:
c) We continue the logical series: .
This expression can be simplified in different ways: open the first bracket using the formula for abbreviated multiplication of the cube of the sum, or factorize the entire expression using the difference of cubes formula. Try to do it yourself using any of the suggested methods.
So, I got the following:
And again let's remember that. This means that we can neglect all terms containing:
We get: .
d) Similar rules can be obtained for large powers:
e) It turns out that this rule can be generalized for a power function with an arbitrary exponent, not even an integer:
| (2) |
The rule can be formulated in the words: “the degree is brought forward as a coefficient, and then reduced by .”
We will prove this rule later (almost at the very end). Now let's look at a few examples. Find the derivative of the functions:
- (in two ways: by formula and using the definition of derivative - by calculating the increment of the function);
- . Believe it or not, this is a power function. If you have questions like “How is this? Where is the degree?”, remember the topic “”!
Yes, yes, the root is also a degree, only fractional: .
This means that our square root is just a power with an exponent:
.
We look for the derivative using the recently learned formula:If at this point it becomes unclear again, repeat the topic “”!!! (about degree with negative indicator)
- . Now the exponent:
And now through the definition (have you forgotten yet?):
;
.
Now, as usual, we neglect the term containing:
. - . Combination of previous cases: .
Trigonometric functions.
Here we will use one fact from higher mathematics:
With expression.
You will learn the proof in the first year of institute (and to get there, you need to pass the Unified State Exam well). Now I’ll just show it graphically:
We see that when the function does not exist - the point on the graph is cut out. But the closer to the value, the closer the function is to. This is what “aims.”
Additionally, you can check this rule using a calculator. Yes, yes, don’t be shy, take a calculator, we’re not at the Unified State Exam yet.
So, let's try: ;
Don't forget to switch your calculator to Radians mode!
etc. We see that the less, the closer value relationship to
a) Consider the function. As usual, let's find its increment:
Let's turn the difference of sines into a product. To do this, we use the formula (remember the topic “”): .
Now the derivative:
Let's make a replacement: . Then for infinitesimal it is also infinitesimal: . The expression for takes the form:
And now we remember that with the expression. And also, what if an infinitesimal quantity can be neglected in the sum (that is, at).
So, we get the following rule: the derivative of the sine is equal to the cosine:
These are basic (“tabular”) derivatives. Here they are in one list:
Later we will add a few more to them, but these are the most important, since they are used most often.
Practice:
- Find the derivative of the function at a point;
- Find the derivative of the function.
Solutions:
- First, let's find the derivative in general view, and then substitute its value:
;
. - Here we have something similar to power function. Let's try to bring her to
normal view:
.
Great, now you can use the formula:
.
. - . Eeeeeee.....What is this????
Okay, you're right, we don't yet know how to find such derivatives. Here we have a combination of several types of functions. To work with them, you need to learn a few more rules:
Exponent and natural logarithm.
There is a function in mathematics whose derivative for any value is equal to the value of the function itself at the same time. It is called “exponent”, and is an exponential function
The basis of this function is a constant - it is infinite decimal, that is, an irrational number (such as). It is called the “Euler number”, which is why it is denoted by a letter.
So, the rule:
Very easy to remember.
Well, let’s not go far, let’s immediately consider the 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: Exhibitor and natural logarithm- functions are uniquely simple in terms of derivatives. 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... Mathematicians call the differential 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 any more in simple form. Therefore, we leave it in this form in the answer.
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 Exam, 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 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.
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.
In other words, a complex function is a function whose argument is another function: .
For the first example, .
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 (we 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.
The content of the article
DERIVATIVE– derivative of the function y = f(x), given on a certain interval ( a, b) at point x of this interval is called the limit to which the ratio of the increment of the function tends f at this point to the corresponding increment of the argument when the increment of the argument tends to zero.
The derivative is usually denoted as follows:
Other designations are also widely used:
Instant speed.
Let the point M moves in a straight line. Distance s moving point, counted from some initial position M 0 , depends on time t, i.e. s there is a function of time t: s= f(t). Let at some point in time t moving point M was at a distance s from the starting position M 0, and at some next moment t+D t found herself in a position M 1 - on distance s+D s from the initial position ( see pic.).
Thus, over a period of time D t distance s changed by the amount D s. In this case they say that during the time interval D t magnitude s received increment D s.
The average speed cannot in all cases accurately characterize the speed of movement of a point M at a point in time t. If, for example, the body at the beginning of the interval D t moved very quickly, and at the end very slowly, then the average speed will not be able to reflect the indicated features of the point’s movement and give an idea of the true speed of its movement at the moment t. To more accurately express the true speed using the average speed, you need to take a shorter period of time D t. Most fully characterizes the speed of movement of a point at the moment t the limit to which the average speed tends at D t® 0. This limit is called the speed of movement in this moment:
Thus, the speed of movement at a given moment is called the limit of the path increment ratio D s to time increment D t, when the time increment tends to zero. Because
Geometric meaning of the derivative. Tangent to the graph of a function.
The construction of tangents is one of those problems that led to the birth of differential calculus. The first published work related to differential calculus, written by Leibniz, was entitled New method maxima and minima, as well as tangents, for which neither fractional nor irrational quantities, and a special type of calculus for this, serve as an obstacle.
Let the curve be the graph of the function y =f(x) in a rectangular coordinate system ( cm. rice.).
At some value x function matters y =f(x). These values x And y the point on the curve corresponds M 0(x, y). If the argument x give increment D x, then the new value of the argument x+D x corresponds to the new function value y+ D y = f(x + D x). The corresponding point of the curve will be the point M 1(x+D x,y+D y). If you draw a secant M 0M 1 and denoted by j the angle formed by a transversal with the positive direction of the axis Ox, it is immediately clear from the figure that .
If now D x tends to zero, then the point M 1 moves along the curve, approaching the point M 0, and angle j changes with D x. At Dx® 0 angle j tends to a certain limit a and the straight line passing through the point M 0 and the component with the positive direction of the x-axis, angle a, will be the desired tangent. Its slope is:
Hence, f´( x) = tga
those. derivative value f´( x) for a given argument value x equals the tangent of the angle formed by the tangent to the graph of the function f(x) at the corresponding point M 0(x,y) with positive axis direction Ox.
Differentiability of functions.
Definition. If the function y = f(x) has a derivative at the point x = x 0, then the function is differentiable at this point.
Continuity of a function having a derivative. Theorem.
If the function y = f(x) is differentiable at some point x = x 0, then it is continuous at this point.
Thus, the function cannot have a derivative at discontinuity points. The opposite conclusion is incorrect, i.e. from the fact that at some point x = x 0 function y = f(x) is continuous does not mean that it is differentiable at this point. For example, the function y = |x| continuous for everyone x(–Ґ x x = 0 has no derivative. At this point there is no tangent to the graph. There is a right tangent and a left one, but they do not coincide.
Some theorems about differentiable functions. Theorem on the roots of the derivative (Rolle's theorem). If the function f(x) is continuous on the segment [a,b], is differentiable at all interior points of this segment and at the ends x = a And x = b goes to zero ( f(a) = f(b) = 0), then inside the segment [ a,b] there is at least one point x= With, a c b, in which the derivative fў( x) goes to zero, i.e. fў( c) = 0.
Finite increment theorem (Lagrange's theorem). If the function f(x) is continuous on the interval [ a, b] and is differentiable at all interior points of this segment, then inside the segment [ a, b] there is at least one point With, a c b that
f(b) – f(a) = fў( c)(b– a).
Theorem on the ratio of the increments of two functions (Cauchy's theorem). If f(x) And g(x) – two functions continuous on the segment [a, b] and differentiable at all interior points of this segment, and gў( x) does not vanish anywhere inside this segment, then inside the segment [ a, b] there is such a point x = With, a c b that
Derivatives of various orders.
Let the function y =f(x) is differentiable on some interval [ a, b]. Derivative values f ў( x), generally speaking, depend on x, i.e. derivative f ў( x) is also a function of x. When differentiating this function, we obtain the so-called second derivative of the function f(x), which is denoted f ўў ( x).
Derivative n- th order of function f(x) is called the (first order) derivative of the derivative n- 1- th and is denoted by the symbol y(n) = (y(n– 1))ў.
Differentials of various orders.
Function differential y = f(x), Where x– independent variable, yes dy = f ў( x)dx, some function from x, but from x only the first factor can depend f ў( x), the second factor ( dx) is the increment of the independent variable x and does not depend on the value of this variable. Because dy there is a function from x, then we can determine the differential of this function. The differential of the differential of a function is called the second differential or second-order differential of this function and is denoted d 2y:
d(dx) = d 2y = f ўў( x)(dx) 2 .
Differential n- of the first order is called the first differential of the differential n- 1- th order:
d n y = d(dn–1y) = f(n)(x)dx(n).
Partial derivative.
If a function depends not on one, but on several arguments x i(i varies from 1 to n,i= 1, 2,… n),f(x 1,x 2,… x n), then in differential calculus the concept of partial derivative is introduced, which characterizes the rate of change of a function of several variables when only one argument changes, for example, x i. Partial derivative of 1st order with respect to x i is defined as an ordinary derivative, and it is assumed that all arguments except x i, keep constant values. For partial derivatives, the notation is introduced
The 1st order partial derivatives defined in this way (as functions of the same arguments) can, in turn, also have partial derivatives, these are second order partial derivatives, etc. Such derivatives taken from different arguments are called mixed. Continuous mixed derivatives of the same order do not depend on the order of differentiation and are equal to each other.
Anna Chugainova
Definition. Let the function \(y = f(x)\) be defined in a certain interval containing the point \(x_0\). Let's give the argument an increment \(\Delta x \) such that it does not leave this interval. Let's find the corresponding increment of the function \(\Delta y \) (when moving from the point \(x_0 \) to the point \(x_0 + \Delta x \)) and compose the relation \(\frac(\Delta y)(\Delta x) \). If there is a limit to this ratio at \(\Delta x \rightarrow 0\), then the specified limit is called derivative of a function\(y=f(x) \) at the point \(x_0 \) and denote \(f"(x_0) \).
$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x_0) $$
The symbol y is often used to denote the derivative. Note that y" = f(x) is a new function, but naturally related to the function y = f(x), defined at all points x at which the above limit exists . This function is called like this: derivative of the function y = f(x).
Geometric meaning derivative is as follows. If it is possible to draw a tangent to the graph of the function y = f(x) at the point with abscissa x=a, which is not parallel to the y-axis, then f(a) expresses the slope of the tangent:
\(k = f"(a)\)
Since \(k = tg(a) \), then the equality \(f"(a) = tan(a) \) is true.
Now let’s interpret the definition of derivative from the point of view of approximate equalities. Let the function \(y = f(x)\) have a derivative at a specific point \(x\):
$$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) = f"(x) $$
This means that near the point x the approximate equality \(\frac(\Delta y)(\Delta x) \approx f"(x)\), i.e. \(\Delta y \approx f"(x) \cdot\Delta x\). The meaningful meaning of the resulting approximate equality is as follows: the increment of the function is “almost proportional” to the increment of the argument, and the coefficient of proportionality is the value of the derivative in given point X. For example, for the function \(y = x^2\) the approximate equality \(\Delta y \approx 2x \cdot \Delta x \) is valid. If we carefully analyze the definition of a derivative, we will find that it contains an algorithm for finding it.
Let's formulate it.
How to find the derivative of the function y = f(x)?
1. Fix the value of \(x\), find \(f(x)\)
2. Give the argument \(x\) an increment \(\Delta x\), go to a new point \(x+ \Delta x \), find \(f(x+ \Delta x) \)
3. Find the increment of the function: \(\Delta y = f(x + \Delta x) - f(x) \)
4. Create the relation \(\frac(\Delta y)(\Delta x) \)
5. Calculate $$ \lim_(\Delta x \to 0) \frac(\Delta y)(\Delta x) $$
This limit is the derivative of the function at point x.
If a function y = f(x) has a derivative at a point x, then it is called differentiable at a point x. The procedure for finding the derivative of the function y = f(x) is called differentiation functions y = f(x).
Let us discuss the following question: how are continuity and differentiability of a function at a point related to each other?
Let the function y = f(x) be differentiable at the point x. Then a tangent can be drawn to the graph of the function at point M(x; f(x)), and, recall, the angular coefficient of the tangent is equal to f "(x). Such a graph cannot “break” at point M, i.e. the function must be continuous at point x.
These were “hands-on” arguments. Let us give a more rigorous reasoning. If the function y = f(x) is differentiable at the point x, then the approximate equality \(\Delta y \approx f"(x) \cdot \Delta x\) holds. If in this equality \(\Delta x \) tends to zero, then \(\Delta y \) will tend to zero, and this is the condition for the continuity of the function at a point.
So, if a function is differentiable at a point x, then it is continuous at that point.
The reverse statement is not true. For example: function y = |x| is continuous everywhere, in particular at the point x = 0, but the tangent to the graph of the function at the “junction point” (0; 0) does not exist. If at some point a tangent cannot be drawn to the graph of a function, then the derivative does not exist at that point.
One more example. The function \(y=\sqrt(x)\) is continuous on the entire number line, including at the point x = 0. And the tangent to the graph of the function exists at any point, including at the point x = 0. But at this point the tangent coincides with the y-axis, i.e., it is perpendicular to the abscissa axis, its equation has the form x = 0. Such a straight line does not have an angle coefficient, which means that \(f"(0)\) does not exist.
So, we got acquainted with a new property of a function - differentiability. How can one conclude from the graph of a function that it is differentiable?
The answer is actually given above. If at some point it is possible to draw a tangent to the graph of a function that is not perpendicular to the abscissa axis, then at this point the function is differentiable. If at some point the tangent to the graph of a function does not exist or it is perpendicular to the abscissa axis, then at this point the function is not differentiable.
Rules of differentiation
The operation of finding the derivative is called differentiation. When performing this operation, you often have to work with quotients, sums, products of functions, as well as “functions of functions,” that is, complex functions. Based on the definition of derivative, we can derive differentiation rules that make this work easier. If C is a constant number and f=f(x), g=g(x) are some differentiable functions, then the following are true differentiation rules:
$$ f"_x(g(x)) = f"_g \cdot g"_x $$
Table of derivatives of some functions
$$ \left(\frac(1)(x) \right) " = -\frac(1)(x^2) $$ $$ (\sqrt(x)) " = \frac(1)(2\ sqrt(x)) $$ $$ \left(x^a \right) " = a x^(a-1) $$ $$ \left(a^x \right) " = a^x \cdot \ln a $$ $$ \left(e^x \right) " = e^x $$ $$ (\ln x)" = \frac(1)(x) $$ $$ (\log_a x)" = \frac (1)(x\ln a) $$ $$ (\sin x)" = \cos x $$ $$ (\cos x)" = -\sin x $$ $$ (\text(tg) x) " = \frac(1)(\cos^2 x) $$ $$ (\text(ctg) x)" = -\frac(1)(\sin^2 x) $$ $$ (\arcsin x) " = \frac(1)(\sqrt(1-x^2)) $$ $$ (\arccos x)" = \frac(-1)(\sqrt(1-x^2)) $$ $$ (\text(arctg) x)" = \frac(1)(1+x^2) $$ $$ (\text(arcctg) x)" = \frac(-1)(1+x^2) $ $(\large\bf Derivative of a function)
Consider the function y=f(x), specified on the interval (a, b). Let x- any fixed point of the interval (a, b), A Δx- an arbitrary number such that the value x+Δx also belongs to the interval (a, b). This number Δx called argument increment.
Definition. Function increment y=f(x) at the point x, corresponding to the argument increment 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 this point to the corresponding increment of the argument Δx
We will call this relation the difference relation. Since the value x we consider fixed, the difference ratio 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 Δx=0, except for the point itself Δx=0. Thus, we have the right to consider the question of the existence of a limit specified function at Δx → 0.
Definition. Derivative of a function y=f(x) at a given fixed point x called the limit at Δx → 0 difference ratio, that is
Provided that this limit exists.
Designation. y′(x) or f′(x).
Geometric meaning of derivative: Derivative of a 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.
Mechanical meaning of derivative: The derivative of the path with respect to time is equal to the speed rectilinear movement points:
Equation of a tangent to a line y=f(x) at the point M 0 (x 0 ,y 0) takes the form
y-y 0 = f′(x 0) (x-x 0).
The normal to a curve at some point is the perpendicular to the tangent at the same point. If f′(x 0)≠ 0, then the equation of the normal to the line y=f(x) at the point M 0 (x 0 ,y 0) is written like this:
The concept of differentiability of a function
Let the function y=f(x) defined over a certain interval (a, b), x- some fixed argument value 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) called differentiable at a given point x, if increment Δy this function at the point x, corresponding to the argument increment Δx, can be represented in the form
Δy = A Δx +αΔx,
Where A- some number independent of Δx, A α - argument function Δx, which is infinitesimal at Δx→ 0.
Since the product of two infinitesimal functions αΔx is infinitesimal more high order, how Δx(property of 3 infinitesimal functions), then we can write:
Δy = A Δx +o(Δx).
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 this point.
Comment. From the continuity of the function y=f(x) at this point x, generally speaking, the differentiability of the function does not follow f(x) at this point. For example, the function y=|x|- continuous at a point x=0, but has no derivative.
Concept of differential function
Definition. Function differential y=f(x) the product of the derivative of this function and the increment of the independent variable is called x:
dy = y′ Δx, df(x) = f′(x) Δx.
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 argument increment Δx, generally speaking, are not equal to each other.
Geometric meaning of differential: The differential of a function is equal to the increment of the ordinate of the tangent to the graph of this function when the argument is incremented Δx.
Rules of differentiation
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 formulas hold:
Consider the 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 y=f(u) And u=φ(x) are differentiable functions of their arguments, then the derivative of a 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 are the functions v=φ(x), u=f(v) And y=F(u)- differentiable functions of their arguments.
Theorem. Let the function y=f(x) increases (or decreases) and is continuous in some neighborhood of the point x 0. Let, in addition, this function be differentiable at the indicated point x 0 and its derivative at this point f′(x 0) ≠ 0. Then in some neighborhood of the corresponding point y 0 =f(x 0) the inverse is defined for y=f(x) function x=f -1 (y), and the indicated inverse function is differentiable at the corresponding point y 0 =f(x 0) and for its derivative at this point y the formula is valid
Derivatives table
Invariance of the form of the first differential
Let's consider the differential of a complex function. If y=f(x), x=φ(t)- functions of their arguments are differentiable, then the derivative of the function y=f(φ(t)) expressed by the formula
y′ t = y′ x x′ t.
A-priory 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 itself is 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 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. However, with an accuracy up to infinity small function higher order of smallness than Δx, the approximate equality is valid
Δy ≈ dy.
The ratio is called the relative error of the equality of this equality. Because Δy-dy=o(Δx), then the relative error of this equality becomes as small as desired with decreasing |Δх|.
Considering that Δy=f(x+δ x)-f(x), dy=f′(x)Δx, we get f(x+δ x)-f(x) ≈ f′(x)Δx or
f(x+δ x) ≈ f(x) + f′(x)Δx.
This approximate equality allows with error o(Δx) replace function f(x) in a small neighborhood of the point x(i.e. for small values Δx) linear function argument Δx, standing on the right side.
Higher order derivatives
Definition. Second derivative (or second order derivative) of a 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) equal to the acceleration of a moving point at the moment of time x.
The third and fourth derivatives are determined similarly.
Definition. n th derivative (or derivative n-th order) functions y=f(x) is 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.
The operation of finding the 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. The first to work in the field of finding derivatives were Isaac Newton (1643-1727) and Gottfried Wilhelm Leibniz (1646-1716).
Therefore, in our time, to find the derivative of any function, you do not need to calculate the above-mentioned limit of the ratio of the increment of the function to the increment of the argument, but you 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 prime sign break down simple functions into components and determine what actions (product, sum, quotient) these functions are related. Further derivatives elementary functions we find in the table of derivatives, and the formulas for the derivatives of the product, sum and quotient are in the rules of differentiation. The derivative table 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 a 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 sine is equal to cosine. We substitute these values into the sum of derivatives and find the derivative required by the condition of the problem:
Example 2. Find the derivative of a function
Solution. We differentiate as a derivative of a sum in which the second term has a constant factor; it can be taken out of the sign of the derivative:
If questions still arise about where something comes from, they are usually cleared up after familiarizing yourself with the table of derivatives and the simplest rules of differentiation. We are moving on 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 equal to 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 for a long time | |
| 3. Derivative of degree. When solving problems, you need to convert non-square roots into powers. | |
| 4. Derivative of a variable to the power -1 | |
| 5. Derivative square root | |
| 6. Derivative of sine | |
| 7. Derivative of cosine |  |
| 8. Derivative of tangent |  |
| 9. Derivative of cotangent |  |
| 10. Derivative of arcsine |  |
| 11. Derivative of arc cosine |  |
| 12. Derivative of arctangent |  |
| 13. Derivative of arc cotangent |  |
| 14. Derivative of the natural logarithm | |
| 15. Derivative of a logarithmic function |  |
| 16. Derivative of the exponent | |
| 17. Derivative of an exponential function |
Rules of differentiation
| 1. Derivative of a sum or difference |  |
| 2. Derivative of the product |  |
| 2a. Derivative of an expression multiplied by a constant factor | |
| 3. Derivative of the quotient |  |
| 4. Derivative of a complex function |  |
Rule 1.If the functions
are differentiable at some point, then the functions are differentiable at the same point
and
those. the derivative of an 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 term, then their derivatives are equal, i.e.
Rule 2.If the functions
are differentiable at some point, then their product is 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.
Corollary 1. The constant factor can be taken out of the sign of the derivative:
Corollary 2. The derivative of the product of several differentiable functions is equal to the sum of the products of the derivative of each factor and all the others.
For example, for three multipliers:
Rule 3.If the functions
differentiable at some point And , then at this point their quotient is also differentiableu/v , and
those. the derivative of the quotient of two functions is equal to a fraction, the numerator of which 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 for things on other pages
When finding the derivative of a product and a quotient in real problems, it is always necessary to apply several differentiation rules at once, so there are more examples on these derivatives in the article"Derivative of the product and quotient of functions".
Comment. You should not confuse a constant (that is, a number) as a term in a 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 typical mistake, which occurs at the initial stage of studying derivatives, but as the average student solves several one- and two-part examples, he no longer makes this mistake.
And if, when differentiating a product or quotient, you have a term u"v, in which 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 (this case is discussed in example 10).
Other common mistake- mechanical solution of the derivative of a complex function as a derivative of a simple function. That's why derivative of a complex function a separate article is devoted. But first we will learn to find derivatives simple functions.
Along the way, you can’t do without transforming expressions. To do this, you may need to open the manual in new windows. Actions with powers and roots And Operations with fractions .
If you are looking for solutions to derivatives of fractions with powers and roots, that is, when the function looks like 
, then follow the lesson “Derivative of sums of fractions with powers and roots.”
If you have a task like 
, then you will take 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 define the parts of the function expression: the entire expression represents a 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 by 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 has 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 turns 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 obtain the following derivative values:
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:
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 the quotient: the derivative of the quotient of two functions is equal to a fraction, the numerator of which 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 us 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 problems in which you need to find the derivative of a function, where there is a continuous pile of roots and powers, such as, for example, 
, then welcome to class "Derivative of sums of fractions with powers and roots" .
If you need to learn more about the derivatives of sines, cosines, tangents and others trigonometric functions, that is, when the function looks like , then a lesson for you "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, the derivative of which we familiarized ourselves with in the table of derivatives. According to the rule of differentiation of the product and table value derivative of the square root we get:
Example 6. Find the derivative of a function
Solution. In this function we see a quotient whose dividend is the square root of the independent variable. Using the rule of differentiation of quotients, which we repeated and applied in example 4, and the tabulated value of the derivative of the square root, we obtain:
To get rid of a fraction in the numerator, multiply the numerator and denominator by .
