When a person has taken the first independent steps in studying mathematical analysis and begins to ask uncomfortable questions, it is no longer so easy to get away with the phrase that “differential calculus was found in cabbage.” Therefore, the time has come to be determined and reveal the secret of the birth tables of derivatives and differentiation rules. Started in the article about the meaning of derivative, which I highly recommend studying, because there we just looked at the concept of a derivative and started clicking on problems on the topic. This same lesson has a pronounced practical orientation, moreover,
the examples discussed below can, in principle, be mastered purely formally (for example, when there is no time/desire to delve into the essence of the derivative). It is also highly desirable (but again not necessary) to be able to find derivatives using the “ordinary” method - at least at the level of two basic lessons: How to find the derivative? and Derivative of a complex function.
But there’s one thing we definitely can’t do without now, it’s function limits. You must UNDERSTAND what a limit is and be able to solve them at least at an intermediate level. And all because the derivative
function at a point is determined by the formula:
Let me remind you of the designations and terms: they call argument increment;
– function increment;
– these are SINGLE symbols (“delta” cannot be “torn off” from “X” or “Y”).
Obviously, what is a “dynamic” variable is a constant and the result of calculating the limit 
– number (sometimes - “plus” or “minus” infinity).
As a point, you can consider ANY value belonging to domain of definition function in which a derivative exists.
Note: the clause "in which the derivative exists" is in general it is significant! So, for example, although a point is included in the domain of definition of a function, its derivative
doesn't exist there. Therefore the formula
not applicable at point
and a shortened formulation without a reservation would be incorrect. Similar facts are true for other functions with “breaks” in the graph, in particular, for arcsine and arccosine.
Thus, after replacing , we get the second working formula:
Pay attention to an insidious circumstance that can confuse the teapot: in this limit, “x”, being itself an independent variable, plays the role of a statistic, and the “dynamics” is again set by the increment. The result of calculating the limit
is the derivative function.
Based on the above, we formulate the conditions of two typical problems:
- Find derivative at a point, using the definition of derivative.
- Find derivative function, using the definition of derivative. This version, according to my observations, is much more common and will be given the main attention.
The fundamental difference between the tasks is that in the first case you need to find the number (optionally, infinity), and in the second –
function In addition, the derivative may not exist at all.
How ?
Create a ratio and calculate the limit.
Where did it come from? table of derivatives and differentiation rules ? Thanks to the only limit
It seems like magic, but
in reality - sleight of hand and no fraud. At the lesson What is a derivative? I began to look at specific examples where, using the definition, I found the derivatives of a linear and quadratic function. For the purpose of cognitive warm-up, we will continue to disturb table of derivatives, honing the algorithm and technical solutions:
Essentially, you need to prove a special case of the derivative of a power function, which usually appears in the table: .
The solution is technically formalized in two ways. Let's start with the first, already familiar approach: the ladder starts with a plank, and the derivative function starts with the derivative at a point.
Consider some (specific) point belonging to domain of definition function in which there is a derivative. Let us set the increment at this point (of course, within the scope o/o -ya) and compose the corresponding increment of the function:
Let's calculate the limit:
The uncertainty 0:0 is eliminated by a standard technique, considered back in the first century BC. Let's multiply
numerator and denominator for the conjugate expression 
:
The technique for solving such a limit is discussed in detail in the introductory lesson. about the limits of functions.
Since you can choose ANY point of the interval as
Then, having made the replacement, we get:
Once again let's rejoice at logarithms:
Find the derivative of a function using the definition of derivative
Solution: Let's consider a different approach to promoting the same task. It is exactly the same, but more rational in terms of design. The idea is to get rid of
subscript and use a letter instead of a letter.
Consider an arbitrary point belonging to domain of definition function (interval), and set the increment in it. But here, by the way, as in most cases, you can do without any reservations, since the logarithmic function is differentiable at any point in the domain of definition.
Then the corresponding increment of the function is:
Let's find the derivative:
The simplicity of the design is balanced by the confusion that can
occur among beginners (and not only). After all, we are used to the fact that the letter “X” changes in the limit! But here everything is different: - an antique statue, and - a living visitor, briskly walking along the corridor of the museum. That is, “x” is “like a constant.”
I will comment on the elimination of uncertainty step by step:
(1)
Using the logarithm property
.
(2) In parentheses, divide the numerator by the denominator term by term.
(3) In the denominator, we artificially multiply and divide by “x” so that
take advantage of the wonderful limit 
, while as infinitesimal acts.
Answer: by definition of a derivative:
Or in short:
I propose to construct two more table formulas yourself:
Find derivative by definition
In this case, it is convenient to immediately reduce the compiled increment to a common denominator. An approximate sample of the assignment at the end of the lesson (first method).
Find derivative by definition
And here everything must be reduced to a remarkable limit. The solution is formalized in the second way.
A number of other tabular derivatives. The complete list can be found in the school textbook, or, for example, the 1st volume of Fichtenholtz. I don’t see much point in copying proofs of differentiation rules from books - they are also generated
formula
Let's move on to actually encountered tasks: Example 5
Find the derivative of a function 
, using the definition of derivative
Solution: use the first design style. Let's consider some point belonging to and set the increment of the argument at it. Then the corresponding increment of the function is:
Perhaps some readers have not yet fully understood the principle by which increments need to be made. Take a point (number) and find the value of the function in it: 
, that is, into the function
instead of "X" you should substitute. Now let's take it
Compiled function increment It can be beneficial to immediately simplify. For what? Facilitate and shorten the solution to a further limit.
We use formulas, open the brackets and reduce everything that can be reduced:
The turkey is gutted, no problem with the roast:
Eventually: 
Since we can choose any real number as a value, we make the replacement and get 
.
Answer : 
a-priory.
For verification purposes, let’s find the derivative using the rules
differentiation and tables:
It is always useful and pleasant to know the correct answer in advance, so it is better to differentiate the proposed function in a “quick” way, either mentally or in a draft, at the very beginning of the solution.
Find the derivative of a function by definition of derivative
This is an example for you to solve on your own. The result is obvious:
Let's go back to style #2: Example 7
Let's find out immediately what should happen. By rule of differentiation of complex functions:
Solution: consider an arbitrary point belonging to, set the increment of the argument at it and make up the increment
Let's find the derivative:
(1) We use the trigonometric formula
(2) Under the sine we open the brackets, under the cosine we present similar terms.
(3) Under the sine we cancel the terms, under the cosine we divide the numerator by the denominator term by term.
(4) Due to the oddness of the sine, we take out the “minus”. Under cosine
we indicate that the term .
(5) We carry out artificial multiplication in the denominator in order to use first wonderful limit. Thus, the uncertainty is eliminated, let’s tidy up the result.
Answer: by definition As you can see, the main difficulty of the problem under consideration rests on
complexity of the very limit + slight originality of packaging. In practice, both methods of design occur, so I describe both approaches in as much detail as possible. They are equivalent, but still, in my subjective impression, it is more advisable for dummies to stick to option 1 with “X-zero”.
Using the definition, find the derivative of the function
This is a task for you to solve on your own. The sample is designed in the same spirit as the previous example.
Let's look at a rarer version of the problem:
Find the derivative of a function at a point using the definition of derivative.
Firstly, what should be the bottom line? Number Let's calculate the answer in the standard way:
Solution: from a clarity point of view, this task is much simpler, since in the formula, instead of
a specific value is considered.
Let us set the increment at the point and compose the corresponding increment of the function:
Let's calculate the derivative at a point:
We use a very rare tangent difference formula 
and once again we reduce the solution to the first one
remarkable limit:
Answer: by definition of derivative at a point.
The problem is not so difficult to solve “in general” - it is enough to replace the nail, or simply depending on the design method. In this case, it is clear that the result will not be a number, but a derived function.
Example 10 Using the definition, find the derivative of the function 
at the point
This is an example for you to solve on your own.
The final bonus task is intended primarily for students with an in-depth study of mathematical analysis, but it will not hurt anyone else either:
Will the function be differentiable? 
at the point?
Solution: It is obvious that a piecewise given function is continuous at a point, but will it be differentiable there?
The solution algorithm, and not only for piecewise functions, is as follows:
1) Find the left-hand derivative at a given point: .
2) Find the right-hand derivative at a given point: .
3) If one-sided derivatives are finite and coincide:
, then the function is differentiable at the point
geometrically, there is a common tangent here (see the theoretical part of the lesson Definition and meaning of derivative).
If two different values are received: 
(one of which may turn out to be infinite), then the function is not differentiable at the point.
If both one-sided derivatives are equal to infinity
(even if they have different signs), then the function is not
is differentiable at the point, but there is an infinite derivative and a common vertical tangent to the graph (see example lesson 5Normal equation) .
In this lesson we will learn to apply formulas and rules of differentiation.
Examples. Find derivatives of functions.
1. y=x 7 +x 5 -x 4 +x 3 -x 2 +x-9. Applying the rule I, formulas 4, 2 and 1. We get:
y’=7x 6 +5x 4 -4x 3 +3x 2 -2x+1.
2. y=3x 6 -2x+5. We solve similarly, using the same formulas and formula 3.
y’=3∙6x 5 -2=18x 5 -2.
Applying the rule I, formulas 3, 5
And 6
And 1.
Applying the rule IV, formulas 5
And 1
.
In the fifth example, according to the rule I the derivative of the sum is equal to the sum of the derivatives, and we just found the derivative of the 1st term (example 4 ), therefore, we will find derivatives 2nd And 3rd terms, and for 1st summand we can immediately write the result.
Let's differentiate 2nd And 3rd terms according to the formula 4
. To do this, we transform the roots of the third and fourth powers in the denominators to powers with negative exponents, and then, according to 4
formula, we find derivatives of powers.
Look at this example and the result. Did you catch the pattern? Fine. This means we have a new formula and can add it to our derivatives table.
Let's solve the sixth example and derive another formula.
Let's use the rule IV and formula 4
. Let's reduce the resulting fractions.
Let's look at this function and its derivative. You, of course, understand the pattern and are ready to name the formula:
Learning new formulas!
Examples.
1. Find the increment of the argument and the increment of the function y= x 2, if the initial value of the argument was equal to 4 , and new - 4,01 .
Solution.
New argument value x=x 0 +Δx. Let's substitute the data: 4.01=4+Δх, hence the increment of the argument Δх=4.01-4=0.01. The increment of a function, by definition, is equal to the difference between the new and previous values of the function, i.e. Δy=f (x 0 +Δx) - f (x 0). Since we have a function y=x2, That Δу=(x 0 +Δx) 2 - (x 0) 2 =(x 0) 2 +2x 0 · Δx+(Δx) 2 - (x 0) 2 =2x 0 · Δx+(Δx) 2 =
2 · 4 · 0,01+(0,01) 2 =0,08+0,0001=0,0801.
Answer: argument increment Δх=0.01; function increment Δу=0,0801.
The function increment could be found differently: Δy=y (x 0 +Δx) -y (x 0)=y(4.01) -y(4)=4.01 2 -4 2 =16.0801-16=0.0801.
2. Find the angle of inclination of the tangent to the graph of the function y=f(x) at the point x 0, If f "(x 0) = 1.
Solution.
The value of the derivative at the point of tangency x 0 and is the value of the tangent of the tangent angle (the geometric meaning of the derivative). We have: f "(x 0) = tanα = 1 → α = 45°, because tg45°=1.
Answer: the tangent to the graph of this function forms an angle with the positive direction of the Ox axis equal to 45°.
3. Derive the formula for the derivative of the function y=xn.
Differentiation is the action of finding the derivative of a function.
When finding derivatives, use formulas that were derived based on the definition of a derivative, in the same way as we derived the formula for the derivative degree: (x n)" = nx n-1.
These are the formulas.
Table of derivatives It will be easier to memorize by pronouncing verbal formulations:
1. The derivative of a constant quantity is zero.
2. X prime is equal to one.
3. The constant factor can be taken out of the sign of the derivative.
4. The derivative of a degree is equal to the product of the exponent of this degree by a degree with the same base, but the exponent is one less.
5. The derivative of a root is equal to one divided by two equal roots.
6. The derivative of one divided by x is equal to minus one divided by x squared.
7. The derivative of the sine is equal to the cosine.
8. The derivative of the cosine is equal to minus sine.
9. The derivative of the tangent is equal to one divided by the square of the cosine.
10. The derivative of the cotangent is equal to minus one divided by the square of the sine.
We teach differentiation rules.
1.
The derivative of an algebraic sum is equal to the algebraic sum of the derivatives of the terms.
2. The derivative of a product is equal to the product of the derivative of the first factor and the second plus the product of the first factor and the derivative of the second.
3. The derivative of “y” divided by “ve” is equal to a fraction in which the numerator is “y prime multiplied by “ve” minus “y multiplied by ve prime”, and the denominator is “ve squared”.
4. A special case of the formula 3.
Let's learn together!
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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 of 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 at a 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) $ $The problem of finding the derivative of a given function is one of the main ones in high school mathematics courses and in higher educational institutions. It is impossible to fully explore a function and construct its graph without taking its derivative. The derivative of a function can be easily found if you know the basic rules of differentiation, as well as the table of derivatives of basic functions. Let's figure out how to find the derivative of a function.
The derivative of a function is the limit of the ratio of the increment of the function to the increment of the argument when the increment of the argument tends to zero.
Understanding this definition is quite difficult, since the concept of a limit is not fully studied in school. But in order to find derivatives of various functions, it is not necessary to understand the definition; let’s leave it to mathematicians and move straight to finding the derivative.
The process of finding the derivative is called differentiation. When we differentiate a function, we will obtain a new function.
To designate them we will use the Latin letters f, g, etc.
There are many different notations for derivatives. We will use a stroke. For example, writing g" means that we will find the derivative of the function g.
Derivatives table
In order to answer the question of how to find the derivative, it is necessary to provide a table of derivatives of the main functions. To calculate the derivatives of elementary functions, it is not necessary to perform complex calculations. It is enough just to look at its value in the table of derivatives.
- (sin x)"=cos x
- (cos x)"= –sin x
- (x n)"=n x n-1
- (e x)"=e x
- (ln x)"=1/x
- (a x)"=a x ln a
- (log a x)"=1/x ln a
- (tg x)"=1/cos 2 x
- (ctg x)"= – 1/sin 2 x
- (arcsin x)"= 1/√(1-x 2)
- (arccos x)"= - 1/√(1-x 2)
- (arctg x)"= 1/(1+x 2)
- (arcctg x)"= - 1/(1+x 2)
Example 1. Find the derivative of the function y=500.
We see that this is a constant. From the table of derivatives it is known that the derivative of a constant is equal to zero (formula 1).
Example 2. Find the derivative of the function y=x 100.
This is a power function whose exponent is 100, and to find its derivative you need to multiply the function by the exponent and reduce it by 1 (formula 3).
(x 100)"=100 x 99
Example 3. Find the derivative of the function y=5 x
This is an exponential function, let's calculate its derivative using formula 4.
Example 4. Find the derivative of the function y= log 4 x
We find the derivative of the logarithm using formula 7.
(log 4 x)"=1/x ln 4
Rules of differentiation
Let's now figure out how to find the derivative of a function if it is not in the table. Most of the functions studied are not elementary, but are combinations of elementary functions using simple operations (addition, subtraction, multiplication, division, and multiplication by a number). To find their derivatives, you need to know the rules of differentiation. Below, the letters f and g denote functions, and C is a constant.
1. The constant coefficient can be taken out of the sign of the derivative
Example 5. Find the derivative of the function y= 6*x 8
We take out a constant factor of 6 and differentiate only x 4. This is a power function, the derivative of which is found using formula 3 of the table of derivatives.
(6*x 8)" = 6*(x 8)"=6*8*x 7 =48* x 7
2. The derivative of a sum is equal to the sum of the derivatives
(f + g)"=f" + g"
Example 6. Find the derivative of the function y= x 100 +sin x
A function is the sum of two functions, the derivatives of which we can find from the table. Since (x 100)"=100 x 99 and (sin x)"=cos x. The derivative of the sum will be equal to the sum of these derivatives:
(x 100 +sin x)"= 100 x 99 +cos x
3. The derivative of the difference is equal to the difference of the derivatives
(f – g)"=f" – g"
Example 7. Find the derivative of the function y= x 100 – cos x
This function is the difference of two functions, the derivatives of which we can also find in the table. Then the derivative of the difference is equal to the difference of the derivatives and don’t forget to change the sign, since (cos x)"= – sin x.
(x 100 – cos x)"= 100 x 99 + sin x
Example 8. Find the derivative of the function y=e x +tg x– x 2.
This function has both a sum and a difference; let’s find the derivatives of each term:
(e x)"=e x, (tg x)"=1/cos 2 x, (x 2)"=2 x. Then the derivative of the original function is equal to:
(e x +tg x– x 2)"= e x +1/cos 2 x –2 x
4. Derivative of the product
(f * g)"=f" * g + f * g"
Example 9. Find the derivative of the function y= cos x *e x
To do this, we first find the derivative of each factor (cos x)"=–sin x and (e x)"=e x. Now let's substitute everything into the product formula. We multiply the derivative of the first function by the second and add the product of the first function by the derivative of the second.
(cos x* e x)"= e x cos x – e x *sin x
5. Derivative of the quotient
(f / g)"= f" * g – f * g"/ g 2
Example 10. Find the derivative of the function y= x 50 /sin x
To find the derivative of a quotient, we first find the derivative of the numerator and denominator separately: (x 50)"=50 x 49 and (sin x)"= cos x. Substituting the derivative of the quotient into the formula, we get:
(x 50 /sin x)"= 50x 49 *sin x – x 50 *cos x/sin 2 x
Derivative of a complex function
A complex function is a function represented by a composition of several functions. There is also a rule for finding the derivative of a complex function:
(u (v))"=u"(v)*v"
Let's figure out how to find the derivative of such a function. Let y= u(v(x)) be a complex function. Let's call the function u external, and v - internal.
For example:
y=sin (x 3) is a complex function.
Then y=sin(t) is an external function
t=x 3 - internal.
Let's try to calculate the derivative of this function. According to the formula, you need to multiply the derivatives of the internal and external functions.
(sin t)"=cos (t) - derivative of the external function (where t=x 3)
(x 3)"=3x 2 - derivative of the internal function
Then (sin (x 3))"= cos (x 3)* 3x 2 is the derivative of a complex function.
Proof and derivation of the formulas for the derivative of the exponential (e to the x power) and the exponential function (a to the x power). Examples of calculating derivatives of e^2x, e^3x and e^nx. Formulas for derivatives of higher orders.
The derivative of an exponent is equal to the exponent itself (the derivative of e to the x power is equal to e to the x power):
(1)
(e x )′ = e x.
The derivative of an exponential function with a base a is equal to the function itself multiplied by the natural logarithm of a:
(2)
.
Derivation of the formula for the derivative of the exponential, e to the x power
An exponential is an exponential function whose base is equal to the number e, which is the following limit:
.
Here it can be either a natural number or a real number. Next, we derive formula (1) for the derivative of the exponential.
Derivation of the exponential derivative formula
Consider the exponential, e to the x power:
y = e x .
This function is defined for everyone. Let's find its derivative with respect to the variable x. By definition, the derivative is the following limit:
(3)
.
Let's transform this expression to reduce it to known mathematical properties and rules. To do this we need the following facts:
A) Exponent property:
(4)
;
B) Property of logarithm:
(5)
;
IN) Continuity of the logarithm and the property of limits for a continuous function:
(6)
.
Here is a function that has a limit and this limit is positive.
G) The meaning of the second remarkable limit:
(7)
.
Let's apply these facts to our limit (3). We use property (4):
;
.
Let's make a substitution. Then ; .
Due to the continuity of the exponential,
.
Therefore, when , . As a result we get:
.
Let's make a substitution. Then . At , . And we have:
.
Let's apply the logarithm property (5):
. Then
.
Let us apply property (6). Since there is a positive limit and the logarithm is continuous, then:
.
Here we also used the second remarkable limit (7). Then
.
Thus, we obtained formula (1) for the derivative of the exponential.
Derivation of the formula for the derivative of an exponential function
Now we derive formula (2) for the derivative of the exponential function with a base of degree a. We believe that and . Then the exponential function
(8)
Defined for everyone.
Let's transform formula (8). For this we will use properties of the exponential function and logarithm.
;
.
So, we transformed formula (8) to the following form:
.
Higher order derivatives of e to the x power
Now let's find derivatives of higher orders. Let's look at the exponent first:
(14)
.
(1)
.
We see that the derivative of function (14) is equal to function (14) itself. Differentiating (1), we obtain derivatives of the second and third order:
;
.
This shows that the nth order derivative is also equal to the original function:
.
Higher order derivatives of the exponential function
Now consider an exponential function with a base of degree a:
.
We found its first-order derivative:
(15)
.
Differentiating (15), we obtain derivatives of the second and third order:
;
.
We see that each differentiation leads to the multiplication of the original function by . Therefore, the nth order derivative has the following form:
.