Answer: Hyperbolic functions - family elementary functions, expressed through an exponent and closely related to trigonometric functions. Hyperbolic functions were introduced by Vincenzo Riccati in 1757 (Opusculorum, Volume I). He obtained them from consideration of the unit hyperbola.
Further research into the properties of hyperbolic functions was carried out by Lambert. Hyperbolic functions are often encountered when calculating various integrals. Some integrals of rational functions and from functions containing radicals are quite simply performed using changes of variables using hyperbolic functions. The derivatives of hyperbolic functions are easy to find because hyperbolic functions are combinations. For example, the hyperbolic sine and cosine are defined as 
The derivatives of these functions have the form 
Hyperbolic functions are given by the following formulas: 1) hyperbolic sine: 
(V foreign literature denoted sinx); 2) hyperbolic cosine: 
(in foreign literature it is designated cosx); 3) hyperbolic tangent: 
(in foreign literature it is designated tanx); 4) hyperbolic cotangent: 
; 5) hyperbolic secant and cosecant: 
Geometric definition: Due to the relationship, hyperbolic functions give a parametric representation of the hyperbola. In this case, the argument is t = 2S, where S is the area of the curvilinear triangle OQR, taken with the “+” sign if the sector lies above the OX axis, and “−” in the opposite case. This definition is similar to the definition of trigonometric functions in terms of the unit circle, which can also be constructed in a similar way. Connection with trigonometric functions: Hyperbolic functions are expressed in terms of trigonometric functions of an imaginary argument. Analytical properties: Hyperbolic sine and hyperbolic cosine are analytic throughout the complex plane, except at the essentially singular point at infinity.
Derivative table.
Answer:
Table of derivatives (which we mainly need): 
46) Derivative of a function – specified parametrically.
Answer:
Let the dependence of two variables x and y on the parameter t be given, varying within the limits from Let the function have an inverse: 
Then we can, taking the composition of functions 
get the dependence of y on x: 
The dependence of the value y on the value x, specified parametrically, can be expressed through the derivatives of the functions since and, according to the formula for the derivative of the inverse function, 
where is the value of the parameter at which the value x we are interested in when calculating the derivative is obtained. Note that applying the formula leads us to the relationship between, again expressed as a parametric relationship: 
the second of these relations is the same one that participated in the parametric specification of the function y(x) . Despite the fact that the derivative is not expressed explicitly, this does not prevent us from solving problems related to finding the derivative by finding the corresponding value of the parameter t. Let's show this with the following example. Example 4.22: Let the dependence between x and y be given parametrically by the following formulas: Find the equation of the tangent to the graph of the dependence y(x) at the point The values are obtained if we take t=1. Let's find the derivatives of x and y with respect to the parameter t: Therefore 
At t=1 we get the value of the derivative; this value specifies slope k of the desired tangent. Coordinates 
the touch points are specified in the problem statement. This means that the tangent equation is as follows: Note that based on the obtained parametric dependence, we can find the second derivative of the function y with respect to the variable x: 
Reference data on hyperbolic functions. Definitions, graphs and properties of hyperbolic sine, cosine, tangent and cotangent. Formulas for sums, differences and products. Derivatives, integrals, series expansions. Expressions through trigonometric functions.
Definitions of hyperbolic functions, their domains of definitions and values
sh x - hyperbolic sine
, -∞ < x < +∞; -∞ < y < +∞ .
ch x - hyperbolic cosine
, -∞ < x < +∞; 1 ≤ y< +∞ .
th x - hyperbolic tangent
, -∞ < x < +∞; - 1 < y < +1 .
cth x - hyperbolic cotangent
X ≠ 0 ; y< -1 или y > +1 .
Graphs of hyperbolic functions
Hyperbolic sine graph y = sh x
Graph of hyperbolic cosine y = ch x
Graph of hyperbolic tangent y = th x
Graph of hyperbolic cotangent y = cth x
Formulas with hyperbolic functions
Relation to trigonometric functions
sin iz = i sh z ; cos iz = ch z
sh iz = i sin z; ch iz = cos z
tg iz = i th z ; cot iz = - i cth z
th iz = i tg z ; cth iz = - i ctg z
Here i is the imaginary unit, i 2 = - 1
.
Applying these formulas to trigonometric functions, we obtain formulas relating hyperbolic functions.
Parity
sh(-x) = - sh x;
ch(-x) = ch x.
th(-x) = - th x;
cth(-x) = - cth x.
Function ch(x)- even. Functions sh(x), th(x), cth(x)- odd.
Difference of squares
ch 2 x - sh 2 x = 1.
Formulas for the sum and difference of arguments
sh(x y) = sh x ch y ch x sh y,
ch(x y) = ch x ch y sh x sh y,
,
,
sh 2 x = 2 sh x ch x,
ch 2 x = ch 2 x + sh 2 x = 2 ch 2 x - 1 = 1 + 2 sh 2 x,
.
Formulas for the products of hyperbolic sine and cosine
,
,
,
,
,
.
Formulas for the sum and difference of hyperbolic functions
,
,
,
,
.
Relation of hyperbolic sine and cosine with tangent and cotangent
,
,
,
.
Derivatives
,
Integrals of sh x, ch x, th x, cth x
,
,
.
Series expansions
sh x
ch x
th x
cth x
Inverse functions
Areasinus
At - ∞< x < ∞
и - ∞ < y < ∞
имеют место формулы:
,
.
Areacosine
At 1 ≤ x< ∞
And 0 ≤ y< ∞
the following formulas apply:
,
.
The second branch of the areacosine is located at 1 ≤ x< ∞
and - ∞< y ≤ 0
:
.
Areatangent
At - 1
< x < 1
and - ∞< y < ∞
имеют место формулы:
,
.
Areacotangent
At - ∞< x < - 1
or 1
< x < ∞
and y ≠ 0
the following formulas apply:
,
.
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
Hyperbolic functions are found in mechanics, electrical engineering and other technical disciplines. Many formulas for hyperbolic functions are similar to formulas for trigonometric functions, except for the property of boundedness.
| № | Function | Name | Derivative  |
| 1. |  | hyperbolic sine |  |
| 2. |  | hyperbolic cosine |  |
| 3. |  | hyperbolic tangent |  |
| 4. |  | hyperbolic cotangent |  |
Formulas for hyperbolic functions
1. 
.
Proof. Let's consider the required difference
. .
Proof. Let's look at the work
.
Let's look at the work 
.
Let's add two products and give similar ones:
Connecting the beginning and the end, we obtain the equality to be proved: .
There are many other properties of hyperbolic functions similar to the properties of trigonometric functions, which are proven in a similar way.
Let us prove formulas for derivatives of hyperbolic functions.
1. Consider the hyperbolic sine 
.
When finding the derivative, we take the constant out of the derivative sign. Next, we apply the property of the derivative of the difference between two functions and . Find the derivative of a function using the table of derivatives: 
. We look for the derivative of the function as the derivative complex function 
.
Therefore, the derivative 
.
Connecting the beginning and the end, we obtain the equality to be proved: 
.
2. Consider the hyperbolic cosine .
We fully apply the previous algorithm, only instead of the property about the derivative of the difference of two functions, we apply the property about the derivative of the sum of these two functions. 
.
Connecting the beginning and the end, we obtain the equality to be proved: 
.
3. Consider the hyperbolic tangent .
We find the derivative using the rule for finding the derivative of a fraction.
4. Derivative of hyperbolic cotangent
can be found as the derivative of a complex function 
.
Connecting the beginning and the end, we obtain the equality to be proved: 
.
Function differential
Let the function 
– is differentiable at the point, then its increment of this function at the point, corresponding to the increment of the argument, can be represented as
where is a certain number independent of , and is a function of the argument , which is infinitesimal for 
.
Thus, the increment of the function is the sum of two infinitesimal terms 
And 
. It was shown that the second term 
is an infinitesimal function of higher order than i.e. (see 8.1). Therefore the first term 
is the principal linear part of the increment of the function . In Remark 8.1. another formula (8.1.1) was obtained for the increment of the function , namely: . (8.1.1)
Definition 8.3.Differential functions at the point is called the main linear part its increment, equal to the product of the derivative 
at this point by an arbitrary increment of argument , and is denoted (or 
):
(8.4)
Function differential also called first order differential.
The differential of an independent variable is understood as any number independent of . Most often, this number is taken to be the increment of the variable, i.e. 
. This is consistent with rule (8.4) for finding the differential of the function
Consider the function 
and find its differential.
Because derivative 
. Thus, we got: 
and differential functions can be found using the formula
. (8.4.1)
Remark 8.7. From formula (8.4.1) it follows that. 
Thus, the notation can be understood not only as a notation for the derivative 
, but also as the ratio of the differentials of the dependent and independent variables.
8.7. Geometric meaning of the differential function
Let the graph of the function 
a tangent is drawn (see Fig. 8.1). Dot 
is on the graph of the function and has an abscissa - . We give an arbitrary increment such that the point 
did not leave the domain of the function definition .
Figure 8.1 Illustration of a graph of a function
The point has coordinates 
. Line segment 
. The point lies on the tangent to the graph of the function and has an abscissa - . From rectangular 
it follows that , where angle is the angle between the positive direction of the axis and the tangent drawn to the graph of the function at point . By definition of the differential of the function and the geometric meaning of the derivative function at point , we conclude that 
. Thus, geometric meaning differential function is that the differential represents the increment of the ordinate of the tangent to the graph of the function at point .
Remark 8.8. Differential and increment for an arbitrary function , generally speaking, are not equal to each other. In the general case, the difference between the increment and differential of a function is infinitesimal of a higher order of smallness than the increment of the argument. From Definition 8.1 it follows that 
, i.e. 
.
In Figure 8.1, the point lies on the graph of the function and has coordinates 
. Line segment .
In Figure 8.1 the inequality is satisfied 
, i.e. 
. But there may be cases when the opposite inequality is true 
. This is done for linear function and for an upwardly convex function.
Definitions of inverse hyperbolic functions and their graphs are given. And also formulas connecting inverse hyperbolic functions - formulas for sums and differences. Expressions through trigonometric functions. Derivatives, integrals, series expansions.
Definitions of inverse hyperbolic functions, their domains of definition and values
arsh x - inverse hyperbolic sine
Inverse hyperbolic sine (areasine), is the inverse function of the hyperbolic sine ( x = sh y) , having a domain of definition -∞< x < +∞ и множество значений -∞ < y < +∞ .
The area sine strictly increases along the entire numerical axis.
arch x - inverse hyperbolic cosine
Inverse hyperbolic cosine (areacosine), is the inverse function of the hyperbolic cosine ( x = сh y) , having a domain of definition 1 ≤ x< +∞ and many meanings 0 ≤ y< +∞ .
The areacosine strictly increases in its domain of definition.
The second branch of the areacosine is also defined for x ≥ 1
and is located symmetrically relative to the abscissa axis, - ∞< y ≤ 0
:
. It strictly decreases in the domain of definition.
arth x - inverse hyperbolic tangent
Inverse hyperbolic tangent (areatangent), is the inverse function of the hyperbolic tangent ( x = th y) , having a domain of definition - 1 < x < 1 and the set of values -∞< y < +∞ .
The areatangent strictly increases in its domain of definition.
arcth x - inverse hyperbolic cotangent
Inverse hyperbolic cotangent (areacotangent), is the inverse function of the hyperbolic cotangent ( x = cth y) , having the domain |x| > 1 and the set of values y ≠ 0 .
The areacotangent strictly decreases in its domain of definition.
Graph of inverse hyperbolic sine (areasine) y = arsh x
Graph of inverse hyperbolic cosine (areacosine) y = arch x , x ≥ 1
The dotted line shows the second branch of the areccosine.
Graph of inverse hyperbolic tangent (areatangent) y = arth x , |x|< 1
Graph of inverse hyperbolic cotangent (areacotangent) y = arcth x , |x| > 1
Formulas with inverse hyperbolic functions
Relation to trigonometric functions
Arsh iz = i Arcsin z;
Arch z = i Arccos z;
Arcsin iz = i Arsh z;
Arccos z = - i Arch z;
Arth iz = i Arctg z;
Arcth iz = - i Arcctg z;
Arctg iz = i Arth z;
Arcctg iz = - i Arcth z;
Here i is the imaginary unit, i 2 = - 1
.
Parity
arsh(-x) = - arsh x;
arch(-x) ≠ arch x;
arth(-x) = - arth x;
arcth(-x) = - arcth x.
Functions arsh(x), arth(x), arcth(x)- odd. Function arch(x)- is not even or odd.
Formulas for connecting inverse hyperbolic sines through tangents and cosines through cotangents
;
;
;
.
Sum and difference formulas
;
;
;
.
Derivatives of inverse hyperbolic functions
;
.
Integrals from arsh x, arch x, arth x, arcth x
arsh x
To calculate the integral of the hyperbolic arcsine, we make the substitution x = sh t and integrate by parts:
.
arch x
Similarly, for the hyperbolic arc cosine. We make the substitution x = ch t and integrate by parts, taking into account that t ≥ 0
:
.
arth x
We make the substitution x = th t and integrate by parts:
;
;
;
.
arcth x
Similarly we get:
.
Series expansions
arsh x
When |x|< 1
arth x
When |x|< 1
the following decomposition takes place:
arcth x
When |x| > 1
the following decomposition takes place:
Inverse functions
Hyperbolic sine
At - ∞< y < ∞
и - ∞ < x < ∞
имеют место формулы:
,
.
Hyperbolic cosine
At 1 ≤ y< ∞
And 0 ≤ x< ∞
the following formulas apply:
,
.
Hyperbolic tangent
At - 1
< y < 1
and - ∞< x < ∞
имеют место формулы:
,
.
Hyperbolic cotangent
At - ∞< y < - 1
or 1
< y < ∞
and x ≠ 0
the following formulas apply:
,
.
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.