Differentiation of Hyperbolic Functions

9 Key excerpts on "Differentiation of Hyperbolic Functions"

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  eBook - PDF

  Single Variable Calculus

  Early Transcendentals, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The derivatives of the hyperbolic functions are easily computed. For example, d dx ssinh xd - d dx S e x 2 e 2x 2 D - e x 1 e 2x 2 - cosh x We list the differentiation formulas for the hyperbolic functions as Table 1. The remain- ing proofs are left as exercises. Note the analogy with the differentiation formulas for trigonometric functions, but note that the signs are different in some cases. 1 Derivatives of Hyperbolic Functions d dx ssinh xd - cosh x d dx scsch xd - 2csch x coth x d dx scosh xd - sinh x d dx ssech xd - 2sech x tanh x d dx stanh xd - sech 2 x d dx scoth xd - 2csch 2 x O y x P(cos t, sin t) ≈+¥=1 Q FIGURE 6 FIGURE 7 0 y x ≈-¥=1 P(cosh t, sinh t) The Gateway Arch in St. Louis was designed using a hyperbolic cosine function (see Exercise 56). Stockbyte/Stockbyte/Getty Images Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 264 CHAPTER 3 Differentiation Rules Any of these differentiation rules can be combined with the Chain Rule, as in the next example. EXAMPLE 2 If y - cosh sx , find dyydx . SOLUTION Using (1) and the Chain Rule, we have dy dx - d dx (cosh sx ) - sinh sx  d dx sx - sinh sx 2 sx ■ ■ Inverse Hyperbolic Functions and Their Derivatives You can see from Figures 1 and 3 that sinh and tanh are one-to-one functions and so they have inverse functions denoted by sinh 21 and tanh 21 . Figure 2 shows that cosh is not one- to-one, but if we restrict the domain to the interval f0, `d, then the function y - cosh x is one-to-one and attains all the values in its range f1, `d.

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  Engineering Mathematics

  A Programmed Approach, 3th Edition

  • C W. Evans, C. Evans(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  Hyperbolic functions 5

  Although we have now explored some of the basic terminology of mathematics and developed the techniques of the differential calculus, we need to pause to extend our algebraic knowledge. In this chapter we shall describe a class of functions known as the hyperbolic functions which are very similar in some ways to the circular functions. We shall use the opportunity to consider in detail what is meant by an inverse function.

  After studying this chapter you should be able to □  Use the hyperbolic functions and their identities; □  Solve algebraic equations which involve hyperbolic functions; □  Differentiate hyperbolic functions; □  Decide when a function has an inverse function; □  Express inverse hyperbolic functions in logarithmic form. We shall also consider a practical problem concerning the sag of a chain. 5.1      DEFINITIONS AND IDENTITIES

  The hyperbolic functions are in some ways very similar to the circular functions. Indeed when we deal with complex numbers (Chapter 10 ) we shall see that there is an algebraic relationship between the two. Initially we shall discuss the hyperbolic functions algebraically, but later we shall see that one of them arises in a physical context.

  The functions cosine and sine are called circular functions because x = cos θ and y = sin θ satisfy the equation x2 + y2 = 1, which is the equation of a circle. The functions known as the hyperbolic cosine (cosh) and the hyperbolic sine (sinh) are called hyperbolic functions because x = cosh u and y = sinh u satisfy the equation x2 – y2 = 1, which is the equation of a rectangular hyperbola.

  We shall define the hyperbolic functions and use these definitions to sketch their graphs. Here then are the definitions:

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  Single Variable Calculus

  Concepts and Contexts, Enhanced Edition

  • James Stewart(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  [Hint: How did we find the derivative of ?] (c) Show that . (d) Use the result of part (c) to find the derivative of . Compare with your answer to part (b). 10. (a) Explain why tanh is a one-to-one function. (b) Find a formula for the derivative of the inverse hyperbolic tangent function . (c) Show that . (d) Use the result of part (c) to find the derivative of . Compare with your answer to part (b). 11. At what point on the curve does the tangent have slope 1? y cosh x tanh 1 x tanh 1 x 1 2 ln 1 x 1 x y tanh 1 x sinh 1 x sinh 1 x ln( x sx 2 1 ) y sin 1 x y sinh 1 x y tanh x y cosh x d dx sinh x cosh x sinh x y sinh x cosh y cosh x sinh y y sinh t x cosh t cosh 2 x sinh 2 x 1 a y a cosh x a y a cosh x a y cosh x y 1 2 e x y 1 2 e x tanh x sinh x cosh x sech x 1 cosh x sinh x e x e x 2 cosh x e x e x 2 e x e x ; Graphing calculator or computer with graphing software required 228 CHAPTER 3 DIFFERENTIATION RULES We know that if , then the derivative can be interpreted as the rate of change of with respect to . In this section we examine some of the applications of this idea to physics, chemistry, biology, economics, and other sciences. Let’s recall from Section 2.6 the basic idea behind rates of change. If changes from to , then the change in is and the corresponding change in is The difference quotient is the average rate of change of y with respect to x over the interval and can be interpreted as the slope of the secant line in Figure 1. Its limit as is the deriv- ative , which can therefore be interpreted as the instantaneous rate of change of y with respect to x or the slope of the tangent line at . Using Leibniz notation, we write the process in the form Whenever the function has a specific interpretation in one of the sciences, its derivative will have a specific interpretation as a rate of change.

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  Mathematics for Engineers and Scientists

  • Alan Jeffrey(Author)
  • 2004(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  The process is called logarithmic differentiation, though it should more properly be called differentiation by means of the logarithmic function. It is best illustrated by example. Suppose we need to find d y/ d x, where y = f (x) g (x) h (x). Taking natural logarithms and using their properties gives ln y = ln f (x) − ln g (x) − ln h (x). Differentiation with respect to x then. gives 1 y d y d x = f ′ (x) f (x) − g ′ (x) g (x) − h ′ (x) h (x), which when multiplied by y gives d y/ d x. The simplification arises from the fact that in general the expression for (1 /y) (d y /d x) is easier to calculate than d y /d x. Example 6.7 Find d y/ d x by means of logarithmic differentiation. if y = (2 x − 7) 5 (3 x + 1) 1 / 2 (2 x − 1) 3 / 2. Solution Taking the natural logarithm gives ln y = 5 ln (2 x − 7) − 1 2 ln (3 x + 1) − 3 2 ln (2 x − 1). Differentiating with respect to x we. find 1 y d y d x = 10 2 x − 7 − 3 2 (3 x + 1) − 3 2 x − 1, which after multiplication by y and simplification becomes d y d x = (72 x 2 + 142 x + 1) (2 x − 7) 4 2 (3 x + 1) 3 / 2 (2 x − 1) 5 / 2. 6.4  Hyperbolic functions It is useful to define new functions called the hyperbolic. sine, written sinhx and pronounced either as ‘cinch x’ or ‘shine x ’, and the hyperbolic cosine, written cosh x and pronounced ‘kosh x ’, which are related to the exponential function. This is achieved as follows. Definition 6.3 For all real x we defme sinh x and cosh x by the requirement that (hyperbolic functions) sinh x = e x − e − x 2, cosh x = e x + e − x 2. It is an immediate consequence of combining the series for. e x and e − x that sinh x = x + x 3 3 ! + x 5 5 ! + x 7 7 ! + … + x 2 n + 1 (2 n + 1) ! + … = ∑ n = 0 ∞ x 2 n + 1 (2 n + 1) !, (6.23) and cosh x = 1 + x 2 2 ! + x 4 4 ! + x 6 6[--=PLGO-SEPARATOR

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  Calculus

  Concepts and Contexts, Enhanced Edition

  • James Stewart(Author)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  4. Prove the identity . ; 5. Graph the curve with parametric equations , . Can you identify this curve? 6. Prove the identity . 7. The identities in Problems 4 and 6 are similar to well-known trigonometric identities. Try to discover other hyperbolic identities by using known trigonometric identities (Reference Page 2) as your inspiration. 8. The differentiation formulas for the hyperbolic functions are analogous to those for the trigonometric functions, but the signs are sometimes different. (a) Show that . (b) Discover formulas for the derivatives of and . 9. (a) Explain why sinh is a one-to-one function. (b) Find a formula for the derivative of the inverse hyperbolic sine function . [ Hint: How did we find the derivative of ?] (c) Show that . (d) Use the result of part (c) to find the derivative of . Compare with your answer to part (b). 10. (a) Explain why tanh is a one-to-one function. (b) Find a formula for the derivative of the inverse hyperbolic tangent function . (c) Show that . (d) Use the result of part (c) to find the derivative of . Compare with your answer to part (b). 11. At what point on the curve does the tangent have slope 1? y cosh x tanh 1 x tanh 1 x 1 2 ln 1 x 1 x y tanh 1 x sinh 1 x sinh 1 x ln ( x s x 2 1 ) y sin 1 x y sinh 1 x y tanh x y cosh x d dx sinh x cosh x sinh x y sinh x cosh y cosh x sinh y y sinh t x cosh t cosh 2 x sinh 2 x 1 a y a cosh x a y a cosh x a y cosh x y 1 2 e x y 1 2 e x tanh x sinh x cosh x sech x 1 cosh x sinh x e x e x 2 cosh x e x e x 2 e x e x ; Graphing calculator or computer with graphing software required Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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  Mathematical Methods

  A Course of Mathematics for Engineers and Scientists

  • Brian H. Chirgwin, Charles Plumpton(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  82 A COURSE OF MATHEMATICS [CH. 2 FIG. 2.2. The graphs of sinh x, cosh x t and tanh x. Division by cosh 2 x gives (iii) sech 2 x = 1 - tanh 2 x. (2.23) Identities involving hyperbolic functions resemble closely corre-sponding identities for trigonometric functions. For example, cosh (*+>>) = Ι^+'+β-*-') = iie^-fe-^.-Ke' + e-') + Ke*-e-*H(e>'-e->'). .·. cosh (x+y) = cosh x cosh j>+sinh x sinh y. Similarly sinh (x+y) = sinh x cosh j>+cosh x sinh y. A hyperbolic identity may be formally obtained from a trigono-metric identity simply by changing sin into sinh and cos into cosh except that wherever the product of two sines occurs in the trigonometric identity a negative sign must be intro-duced in the hyperbolic identity. This result follows directly from eqns. (7.9). § 2 : 4 ] THE TECHNIQUE OF DIFFERENTIATION 83 Example. (i) cos 5x = coss x - 10 cos3 x sinax+ 5 cos x sin4x. :. cosh 5x = COShsiX- 10cosh3 x( - 1)sinh'x+ 5 cosh x( - 1)' sinh'x. : . cosh 5x = cosV x+ 10 cosh3x sinhx x +5 cosh x sinh' x. sin 5x = 5 sin x- 20 sin3 x+ 16 sins x.f :. sinh 5x = 5 sinh x-20(- 1) sinh3 x+ la(- 1)' sinh' x. :. sinh 5x = 5 sinh x+ 20 sinh3x+ 16 sinhs x. (ii) The derivatives of the hyperbolic functions follow at once from the definitions, viz. f (XI fw sinh x cosh x cosh x sinh x tanh x sech2 x sech x -sechxtanhx coth x - cosech2 x cosech x - COSeCh x ~ 0 t h x Exercises 2 : 4 Differentiate the following functions w.r. to x: e.Z' 1. ez'. 2. -. 3. e.'nz. 4. e*sin bx. 5. sinh*xcosh x. (ei-l) 7. tanh x-- tanh* x. 8. In sin x. 9. In cos x. (ez+ 1) * X 1 3 6. - 10. In (=). 1 +x X 16. In [ d(x.+l)-x]. 17. tan (a+bz). 18. xzlnz. (9 + 1): ( ~ -1); ' 19. In . 20. ~ 21. Solve the equation sinh x = e-'+ 1. 22. Express tanh x in terms of e- and show directly from the definitions of the hvmrbolic functions that

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  Anton's Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  129 3 We begin this chapter by extending the process of differentiation to functions that are either difficult or impossible to differentiate directly. Next we will develop a number of new derivative formulas that include the derivatives of logarithmic, exponential, and inverse trigonometric functions. Later in the chapter, we will consider some applications of the derivative. These will include some new methods for finding rates of change as well as the use of linear functions to approximate nonlinear functions. Finally, we will discuss L’H ˆ opital’s rule, a powerful tool for evaluating limits. TOPICS IN DIFFERENTIATION 3.1 IMPLICIT DIFFERENTIATION In this section we will describe a method for differentiating functions that are not expressed in the form y = f (x). FUNCTIONS DEFINED EXPLICITLY AND IMPLICITLY An equation of the form y = f (x) is said to define y explicitly as a function of x. However, sometimes functions are defined by equations not of this form. For example, the equation yx + y + 1 = x (1) defines y as a function of x since it can be rewritten as y = x − 1 x + 1 We say that (1) defines y implicitly as a function of x, the function being f (x) = x − 1 x + 1 x y x y x 2 + y 2 = 1 y = √1 − x 2 x y y = − √ 1 − x 2 Figure 3.1.1 An equation in x and y can implicitly define more than one function of x. For example, if we solve the equation of the circle x 2 + y 2 = 1 (2) for y in terms of x, we obtain y = ± √ 1 − x 2 , so we have found two functions that are defined implicitly by (2), namely, f 1 (x) = √ 1 − x 2 and f 2 (x) = − √ 1 − x 2 (3) The graphs of these functions are the upper and lower semicircles of the circle x 2 + y 2 = 1 (Figure 3.1.1). This leads us to the following definition. 3.1.1 definition We will say that a given equation in x and y defines the function f implicitly if the graph of y = f (x) coincides with a portion of the graph of the equation.

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  A Course of Mathematics for Engineers and Scientists

  Volume 1

  • Brian H. Chirgwin, Charles Plumpton(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  The graphs of sinh x y cosh x and tanh a; are shown in Fig. 10. From the formal definitions given above we derive at once the identities (i) cosh x ± sinh x = e ±x , (2.21) (ii) cosh 2 a; — sinh 2 a; = (cosh x — sinh x) (cosh x + sinh x) = 1. .·. cosh 2 a; = 1 + sinh 2 a (2.22) § 2 : 4 THE TECHNIQUE OF DIFFERENTIATION 45 Division by cosh 2 x gives (iii) sech 2 z = 1 — tanh 2 x. (2.23) Identities involving hyperbolic functions resemble closely corresponding identities for trigonometric functions. For example cosh(x + y) = (e x+ y + e~ x ~v) = i (e* + e*) · | ( e ^ + e~v) + + ^(e* — e-*) . Ke» — e~v). .·. cosh (x + y) = cosh x cosh 2/ + sinh # sinh 2/. Similarly sinh (a; + y) = sinh # cosh z/ + cosh # sinh 2/. A hyperbolic identity may be formally obtained from a trigonometric identity simply by changing 'sin' into 'sinh' and 'cos' into 'cosh' except FIG. 10. The graphs of sinh x, cosh x, and tanh x. that wherever the product of two sines occurs in the trigonometric identity a negative sign must be introduced in the hyperbolic identity. This result follows directly from equations (7.6). 46 A COURSE OF MATHEMATICS Examples. (i) cos 5x — cos 5 a; — 10 cos 3 a; sin 2 a; + 5 cos x sin 4 a;. .·. cosh 5x — cosh 5 a: —-10 cosh 3 a;(— 1) sinh 2 a; + 5 cosh x(— l) 2 sinh 4 a;· .·. cosh 5x = cosh 5 a; + 10 cosh 3 a: sinh 2 a; + 5 cosh x sinh 4 a;. (ii) sin 5x = 5 sin x — 20 sin 3 a; + 16 sin 5 a;. .·. sinh 5x = 5 sinh x — 20(— 1) sinh 3 a; + 16(— l) 2 sinh 5 a:. .·. sinh 5x = 5 sinh x + 20 sinh 3 a; + 16 sinh 5 a;. The derivatives of the hyperbolic functions follow at once from the definitions, viz. f(x) f{x) sinh x cosh x cosh x sinh x tanh x sech 2 :r coth x — cosech 2 # sech x — sech x tanh x cosech x — cosech x coth x Exercises 2:4 Differentiate the following functions w. r. to x : e ax 2 1. e x * . 2. . 3. e 8lniC . 4. e ax sinbx. 5. sinh n a; cosh m a;. x 7. tanh x — tanh 2 x. 8. log sin x . 9. log cos a;. 10. log o 6. ( e * -l ) (e* + 1) 1 +x 1 — x)' 11.

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  A Course of Mathematical Analysis

  International Series of Monographs on Pure and Applied Mathematics

  • A. F. Bermant, I. N. Sneddon, S. Ulam, M. Stark(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  180 COURSE OF MATHEMATICAL ANALYSIS If we take the rectangular hyperbola a = b = 1, the parameter t can be given the following geometrical meaning: it is equal to twice the area of the curvilinear triangle bounded by the real semi-axis, the straight segment joining the centre to the given point of the hyperbola, and the hyperbola itself (see Sec. 117). The parameter t has a precisely similar meaning in the equations of a circle of radius 1: it is equal to twice the area of the sector of the circle, bounded by the radii joining the centre to the initial point (1,0) and to the given point of the circle. Because of this fact, the functions cosh t and sinh t (as also tanh t, etc.) were called the hyperbolic functions; whilst the functions cos£ and sin£ (and the other trigonometric functions) are also called circular. (4) The cycloid is the curve described by a given point of a circle which rolls without slipping* along a straight line (Fig. 65). The straight line along which the rolling occurs is called the base line, whilst the rolling circle is the generating circle. If the base line is taken as Ox, and we take as origin the point of the line which coincides with the fixed point M 0 of the generating circle, the equation of the cycloid has the form x ± y^/(2a — y) = aarccos , 0 ^χ^2πα, a where a is the radius of the circle. The parametric equations of the cycloid are very simple: x = a(t — sin£), y = a(l — cosi). * One curve is said to roll along another without slipping if the length of arc through which the moving curve rotates is exactly equal to the length of the piece of fixed curve which is traversed. FIG. 6 5 DERIVATIVES AND DIFFERENTIALS 181 In fact, we take as parameter the angle t, through which the circle rotates when the fixed point passes from M 0 to the position M. This angle is equal to the angle between the radius of the rolling circle drawn to the point M, and the radius drawn to the point P at which it touches Ox.

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