3.11 hyperbolic functions 1 dr. erickson. certain even and odd combinations of the exponential...

3.11 Hyperbolic Functions1

Chapter 3 – Differentiation Rules

3.11 Hyperbolic Functions

Dr. Erickson


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.

Definition of the Hyperbolic Functions

sinh2

cosh2

sinhtanh

cosh

x x

x x

x x

x x

e ex

e ex

x e ex

x e e

1 2csch

sinh1 2

sech cosh

coshcoth

sinh

x x

x x

x x

x x

xx e e

xx e e

x e ex

x e e

Dr. Erickson


The hyperbolic functions satisfy a number of identities that are similar to well-known trig identities. Here are some of those hyperbolic identities.

Hyperbolic Identities

2 2 2 2

sinh( ) sinh cosh( ) cosh

cosh sinh 1 1- tanh sec h

sinh( ) sinh cosh cosh sinh

cosh( ) cosh cosh sinh s

x x x x

x x x x

x y x y x y

x y x y x

inh y

Dr. Erickson


Note the analogy with the differentiation formulas for trig functions, but be aware that the signs are different in some cases.

Derivatives of Hyperbolic Functions

2

sinh cosh

cosh sinh

tanh sech

dx x

dxd

x xdxd

x xdx

2

csch csch coth

sech sech tanh

coth csch

dx x x

dxd

x x xdxd

x xdx

Dr. Erickson


The sinh and tanh are one-to-one functions and so they have inverse functions denoted by sinh–1 and tanh–1. The cosh is not one-to-one, but when restricted to the domain [0, ) it becomes one-to-one.

The inverse hyperbolic cosine function is defined as the inverse of this restricted function.

Inverse Hyperbolic Functions

Dr. Erickson


We can sketch the graphs of sinh–1, cosh–1, and tanh–1


domain = range =

domain = [1, ) range = [0, )

domain = (–1, 1) range =

Dr. Erickson


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.


1 2

1 2

1

sinh ln 1

cosh ln 1 1

1 1tanh ln 1 1

2 1

x x x x

x x x x

xx x

x

Dr. Erickson


Notice that the derivatives of tanh-1x and coth-1x appear to be identical. But, the domains of these functions have no numbers in common.

tanh-1x is defined for |x|<1 and coth-1x is defined for |x|>1

Derivatives of Inverse Hyperbolic Functions

1

2

1

2

12

1sinh

11

cosh1

1tanh

1

dx

dx xd

xdx xd

xdx x

1

2

1

2

12

1csch

1

1sech

11

coth1

dx

dx x x

dx

dx x xd

xdx x

Dr. Erickson


Find the numerical value of each expression.

Book Example 1- pg. 262 # 3

a. sinh(ln 2)

b. sinh 2

Dr. Erickson


If , find the values of the other

hyperbolic functions at x.

Note: Must use hyperbolic identities not trig identities.

Book Example 2 – pg. 262 # 20

12tanh

13x

Dr. Erickson


The Gateway Arch in St. Louis was designed by Eero Saarinen and constructed using the equation for the central curve of the arch, where x and y are measured in meters and |x|91.20.a) Graph the central curve.

b) What is the height of the arch at its center?

c) At what points is the height of the arch 100 m?

d) What is the slope of the arch at the points in part c?

Example 3

211.49 20.96cosh 0.03291765y x

Dr. Erickson


Find the derivative of the following and simplify where possible.

Example 4

2

2

1. ( ) tanh 1

2. ( ) cosh(ln )

3. ( ) coth 1

4. ( ) csch 1 ln csch

xf x e

g x x

h x x x

f t t t

2 1

1 2

5. ( ) sinh cosh

6. ( ) sinh (2 )

7. ( ) tanh ln 1

j x x

k x x x

m x x x x

Dr. Erickson

3.11 hyperbolic functions 1 dr. erickson. certain even and odd combinations of the exponential...

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