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DIFFERENTIATION RULES

3

Certain even and odd combinations of

the exponential functions ex and e-x arise so

frequently in mathematics and its applications

that they deserve to be given special names.


In many ways, they are analogous to

the trigonometric functions, and they have

the same relationship to the hyperbola that

the trigonometric functions have to the circle.

For this reason, they are collectively called

hyperbolic functions and individually called

hyperbolic sine, hyperbolic cosine, and so on.


3.11

Hyperbolic Functions

In this section, we will learn about:

Hyperbolic functions and their derivatives.


1sinh csc h

2 sinh

1cosh sec h

2 cosh

sinh coshtanh coth

cosh sinh

x x

x x

e ex x

x

e ex x

x

x xx x

x x

DEFINITION

The graphs of hyperbolic sine and cosine

can be sketched using graphical addition,

as in these figures.

HYPERBOLIC FUNCTIONS

Note that sinh has domain and

range , whereas cosh has domain

and range .[1, )


The graph of tanh is shown.

It has the horizontal asymptotes y = 1.


Some mathematical uses of hyperbolic

functions will be seen in Chapter 7.

Applications to science and engineering

occur whenever an entity such as light,

velocity, or electricity is gradually absorbed

or extinguished. The decay can be represented by hyperbolic functions.

APPLICATIONS

The most famous application is

the use of hyperbolic cosine to describe

the shape of a hanging wire.

APPLICATIONS

It can be proved that, if a heavy flexible cable

is suspended between two points at the same

height, it takes the shape of a curve with

equation y = c + a cosh(x/a) called a catenary.

The Latin word

catena means

‘chain.’

APPLICATIONS

Another application occurs in the

description of ocean waves. The velocity of a water wave with length L moving

across a body of water with depth d is modeled by

the function

where g is the acceleration due to gravity.

2tanh

2

gL dv

L

APPLICATIONS

The hyperbolic functions satisfy

a number of identities that are similar to

well-known trigonometric identities.

HYPERBOLIC IDENTITIES

2 2

2 2

sinh( ) sinh

cosh( ) cosh

cosh sinh 1

1 tanh sech

sinh( ) sinh cosh cosh sinh

cosh( ) cosh cosh sinh sinh

x x

x x

x x

x x

x y x y x y

x y x y x y

HYPERBOLIC IDENTITIES

We list some identities here.

Prove:

a. cosh2x – sinh2x = 1

b. 1 – tanh2 x = sech2x

HYPERBOLIC FUNCTIONS Example 1

2 2

2 2

2 2 2 2

cosh sinh2 2

2 2

4 4

41

4

x x x x

x x x x

e e e ex x

e e e e

HYPERBOLIC FUNCTIONS Example 1 a

We start with the identity proved in (a):

cosh2x – sinh2x = 1

If we divide both sides by cosh2x, we get:

2

2 2

2 2

sinh 11

cosh cosh

or 1 tanh sech

x

x x

x x

HYPERBOLIC FUNCTIONS Example 1 b

The identity proved in Example 1 a

gives a clue to the reason for the name

‘hyperbolic’ functions, as follows.


If t is any real number, then the point

P(cos t, sin t) lies on the unit circle x2 + y2 = 1

because cos2 t + sin2 t = 1.

In fact, t can be

interpreted as the radian

measure of

in the figure.

POQ


For this reason, the trigonometric

functions are sometimes called

circular functions.


Likewise, if t is any real number, then

the point P(cosh t, sinh t) lies on the right

branch of the hyperbola x2 - y2 = 1 because

cosh2 t - sin2 t = 1 and cosh t ≥ 1.

This time, t does not represent the measure

of an angle.


However, it turns out that t represents twice

the area of the shaded hyperbolic sector in

the first figure. This is just as in the trigonometric case t represents

twice the area of the shaded circular sector in the

second figure.


The derivatives of the hyperbolic

functions are easily computed.

For example,

(sinh ) cosh2 2

x x x xd d e e e ex x

dx dx

DERIVATIVES OF HYPERBOLIC FUNCTIONS

We list the differentiation formulas for

the hyperbolic functions here.

2 2

(sinh ) cosh (csc h ) csc h coth

(cosh ) sinh (sec h ) sec h tanh

(tanh ) sec h (coth ) csc h

d dx x x x x

dx dx

d dx x x x x

dx dx

d dx x x x

dx dx

DERIVATIVES Table 1

DERIVATIVES

Note the analogy with the differentiation

formulas for trigonometric functions. However, beware that the signs are different

in some cases.

2 2

(sinh ) cosh (csc h ) csc h coth

(cosh ) sinh (sec h ) sec h tanh

(tanh ) sec h (coth ) csc h

d dx x x x x

dx dx

d dx x x x x

dx dx

d dx x x x

dx dx

Any of these differentiation rules can

be combined with the Chain Rule.

For instance,

sinh(cosh ) sinh

2

d d xx x x

dx dx x

Example 2DERIVATIVES

You can see from the figures that sinh

and tanh are one-to-one functions.

So, they have inverse functions denoted by

sinh-1 and tanh-1.

INVERSE HYPERBOLIC FUNCTIONS

This figure shows that cosh is not

one-to-one.

However, when restricted to the domain

[0, ∞], it becomes one-to-one.

INVERSE FUNCTIONS

The inverse hyperbolic cosine

function is defined as the inverse

of this restricted function.

INVERSE FUNCTIONS

1

1

1

sinh sinh

cosh cosh and 0

tanh tanh

y x y x

y x y x y

y x y x

Definition 2INVERSE FUNCTIONS

The remaining inverse hyperbolic functions

are defined similarly.

By using these figures,

we can sketch the graphs

of sinh-1, cosh-1, and

tanh-1.

INVERSE FUNCTIONS

The graphs of sinh-1,

cosh-1, and tanh-1 are

displayed.

INVERSE FUNCTIONS

Since the hyperbolic functions are defined

in terms of exponential functions, it’s not

surprising to learn that the inverse hyperbolic

functions can be expressed in terms of

logarithms.

INVERSE FUNCTIONS

In particular, we have:

1 2

1 2

1 12

sinh ln 1

cosh ln 1 1

1tanh ln 1 1

1

x x x x

x x x x

xx x

x

INVERSE FUNCTIONS Defns. 3, 4, and 5

Show that .

Let y = sinh-1 x. Then,

So, ey – 2x – e-y = 0

Or, multiplying by ey, e2y – 2xey – 1 = 0

This is really a quadratic equation in ey:

(ey)2 – 2x(ey) – 1 = 0

1 2sinh ln 1x x x

sinh2

y ye ex y

INVERSE FUNCTIONS Example 3

Solving by the quadratic formula,

we get:

Note that ey > 0, but

(because ).

So, the minus sign is inadmissible and we have:

Thus,

222 4 4

12

y x xe x x

2 1 0x x2 1x x

2 1ye x x2ln( ) ln 1yy e x x

INVERSE FUNCTIONS Example 3

1 1

2 2

1 1

2 2

1 1

2 2

1 1(sinh ) (csc h )

1 1

1 1cosh (sec h )

1 1

1 1(tanh ) (coth )

1 1

d dx x

dx dxx x x

d dx x

dx dxx x x

d dx x

dx dxx x

DERIVATIVES Table 6

The inverse hyperbolic functions are

all differentiable because the hyperbolic

functions are differentiable.

The formulas in Table 6 can be proved either by

the method for inverse functions or by differentiating

Formulas 3, 4, and 5.

DERIVATIVES

Prove that .

Let y = sinh-1 x. Then, sinh y = x.

If we differentiate this equation implicitly

with respect to x, we get:

As cosh2 y - sin2 y = 1 and cosh y ≥ 0, we have:

So,

1

2

1(sinh )

1

dx

dx x

cosh 1dy

ydx

2cosh 1 sinhy y

2 2

1 1 1

cosh 1 sinh 1

dy

dx y y x

DERIVATIVES E. g. 4—Solution 1

From Equation 3, we have:

1 2

2

2

2 2

2

22 2

sinh ln 1

11

1

11

1 1

1 1

11 1

d dx x x

dx dx

dx x

dxx x

x

x x x

x x

xx x x

DERIVATIVES E. g. 4—Solution 2

Find .

Using Table 6 and the Chain Rule,

we have:

1tanh (sin )d

xdx

1

2

2

2

1tanh (sin ) (sin )

1 (sin )

1cos

1 sin

cossec

cos

d dx x

dx x dx

xx

xx

x

DERIVATIVES Example 5

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