Chapter 2: Differentiation
Section 2.9: The Hyperbolic Functions and Their Derivatives
Essentials
Introduction
Table 2.9.1 lists the definitions of the six hyperbolic functions and their derivatives. As in Section 2.6, there are two cases where Maple's derivatives differ from what is found in the typical calculus text.
Hyperbolic Functions
Maple's Derivatives
Textbook Derivatives
sinhx≡ex−e−x2
ⅆⅆ x sinhx = coshx
coshx
coshx≡ex+e−x2
ⅆⅆ x coshx = sinhx
sinhx
tanhx≡sinhxcoshx=ex−e−xex+e−x
ⅆⅆ x tanhx = 1−tanhx2
sech2x
cothx≡coshxsinhx=ex+e−xex−e−x
ⅆⅆ x cothx = 1−cothx2
−csch2x
sechx≡1coshx=2ex+e−x
ⅆⅆ x sechx = −sechxtanhx
−sechx tanhx
cschx≡1sinhx=2ex−e−x
ⅆⅆ x cschx = −cschxcothx
−cschx cothx
Table 2.9.1 Definitions and derivatives of the hyperbolic functions
As for the derivatives of the trig functions, Maple returns the derivatives of the hyperbolic tangent and cotangent functions in a form that differs from the form found in the typical calculus textbook. (Note the red cells in Table 2.9.1.) This is because early-on, the Maple programmers opined that for these two functions, returning the derivative in terms of the same function was somehow "simpler" than returning it in terms of a different function. Table 2.9.2 contains Maple code for modifying the differentiation rules for these two functions. (The restart is in deference to any remember-table issues.)
restart; unprotect(`diff/tanh`): unprotect(`diff(coth`): `diff/tan` := proc(a,x) sech(a)^2*diff(a,x);end: `diff/cot` := proc(a,x) -csch(a)^2*diff(a,x);end:
Table 2.9.2 Code to modify the differentiation rules for hyperbolic tangent and cotangent
If the differentiation rules are arranged as in Table 2.9.3, certain relations between all the hyperbolic functions and their derivatives can be observed.
ddx sinhx=coshx
ddx coshx= sinhx
ddx tanhx=sech2x
ddx cothx=−csch2x
ddx sechx= −sechxtanhx
ddx cschx= −cschxcothx
Table 2.9.3 Derivatives of the hyperbolic functions
If each function in the left-hand column is replaced by its co-function, and a minus sign inserted for just the derivative of the hyperbolic cotangent, the right-hand column results.
Table 2.9.4 compares the derivatives of the trig and hyperbolic functions.
Function
Trigonometric
Hyperbolic
sine
cosx
cosine
−sinx
tangent
sec2x
cotangent
−csc2x
secant
secx tanx
cosecant
−cscx cotx
Table 2.9.4 Comparison of derivatives for trig and hyperbolic functions
Graphs of the Hyperbolic Functions
Table 2.9.5 contains graphs of the six hyperbolic functions. The graphs in the second row are graphs of the reciprocals of the functions graphed in the first row. Thus, the graph below that of sinhx is the graph of cschx.
Table 2.9.5 Graphs of the six hyperbolic functions
From their graphs, infer that sinhx is an odd function; and coshx, and even function. Note also that coshx≥1 for all real x.
Identities for the Hyperbolic Functions
Table 2.9.6 lists some identities for the hyperbolic functions, and compares them to their trigonometric counterparts.
Identity
Pythagorean
cos2x+sin2x=1
1+tan2x=sec2x
1+ cot2x=csc2x
cosh2x−sinh2x=1
1−tanh2x=sech2x
1− coth2x= −csch2x
Addition
sinx±y=sinxcosy±cosxsiny
cosx±y=cosxcosy∓sinxsiny
sinhx±y=sinhxcoshy±coshxsinhy
coshx±y=coshxcoshy±sinhxsinhy
Double-angle
sin2 x=2 sinxcosy
cos2 x=cos2x−sin2x cos2 x=2 cos2x−1
cos2 x=1−2 sin2x
sinh2 x=2 sinhxcoshy
cosh2 x=cosh2x+sinh2x
cosh2 x=2 cosh2x−1
cosh2 x=1+2 sinh2x
Half-angle
sinx2= ±1−cosx2
cosx2= ±1+cosx2
sinhx2= ±coshx−12
coshx2=coshx+12
Table 2.9.6 Some identities for the trigonometric and hyperbolic functions
Why the Name Hyperbolic?
Figure 2.9.1 suggests why the trig functions are also called the circular functions. Similarly, Figure 2.9.2 suggests why the hyperbolic functions have their name.
p1:=plot([cos(t),sin(t),t=0..2*Pi]): p2:=plot([[cos(Pi/3),sin(Pi/3)]],style=point, symbol=solidcircle,symbolsize=15,color=black): p3:=plots:-textplot([.8,.88,typeset(``(cos(t),sin(t)))]): plots:-display(p1,p2,p3,labels=[x,y]);
Figure 2.9.1 Graph of x2+y2=1
p4:=plots:-implicitplot(x^2-y^2=1,x=-3..3,y=-3..3): p5:=plot([[2,sqrt(3)]],style=point, symbol=solidcircle,symbolsize=15,color=black): p6:=plots:-textplot([2.1,1.75,typeset(``(cosh(t),sinh(t)))],align=right): plots:-display(p4,p5,p6);
Figure 2.9.2 Graph of x2−y2=1
The circular functions can be defined in terms of the sine and cosine functions, which themselves satisfy the identity cos2t+sin2t=1. The parametric curve defined by x=cost,y=sint is therefore the unit circle.
Similarly, the hyperbolic functions can be defined in terms of the hyperbolic sine and cosine functions, which themselves satisfy the identity cosh2x−sinh2x=1. The parametric curve defined by x=cosht,y=sinht is therefore the hyperbola shown in Figure 2.9.2.
Précis
The six hyperbolic functions are defined in Table 2.9.1.
The derivatives of the six hyperbolic functions are stated in Table 2.9.3.
Examples
Example 2.9.1
Verify the differentiation rule for sinhx.
Example 2.9.2
Verify the differentiation rule for tanhx.
Example 2.9.3
Verify the first hyperbolic Pythagorean identity in Table 2.9.5.
Example 2.9.4
It can be shown that for a sufficiently idealized wire cable of length c, hanging between two supports at x1,y1 and x2,y2 in a vertical xy-plane, the equation describing the shape of the cable is of the form yx=k coshd+x/k−λ, with c constrained by the equation c=k sinhd+x2/k−sinhd+x1/k. In this context, the curve is called a catenary, from the Latin catina (chain). If such a cable of length c=2 hangs between the points 0,1 and 1,3/2, find the equation of the resulting catenary, and draw its graph.
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