Hyperbolic Functions

HYPERBOLIC FUNCTIONS

Certain combinations of exponential functions ex and e-x are called hyperbolic functions. They occur frequently in mathematical and engineering applications. We have

The odd combination is called hyperbolic sine of x and the even combination is called hyperbolic cosine of x using these two functions we define other hyperbolic functions.

Definition 2.19

The prefix hyperbolic is due to the fact that (cos hθ, sin hθ) is a point on the hyperbola x2 - y2 = 1, much the same way as

(cos θ, sin θ ) is a point on the circle x2 - y2 = 1.

The hyperbolic functions possess many properties that resemble those of trigonometric functions.

1. Hyperbolic Identities

These identities can be verified using the definitions

2. Derivatives of Hyperbolic Functions

Proof: 

Similarly, we can prove others

3. Inverse Hyperbolic Functions

We saw inverse trigonometric functions exist in the restricted domains.

Similarly, inverse hyperbolic functions exist

Since hyperbolic functions are defined in terms of ex, the inverse hyperbolic functions can be expressed in terms of natural logarithms.

Proof: Let y = sinh -1 x. Then sin y = x

Similarly, we can prove the other inverse hyperbolic functions

4. Derivatives of Inverse Hyperbolic Functions

Proof: 

Similarly, we can prove the following

Formulae

Derivatives of hyperbolic functions

Derivative of inverse hyperbolic functions

WORKED EXAMPLES

Example 1 

Find the derivative of f(x) = tanh(1 + e2x)

Solution 

Example 2 

Find derivative of tan h-1(sin x)

Solution 

Let y = tan h-1(sin x)

Example 3 

Solution 

Example 4 

Solution 

Given

EXERCISE 2.9

I. Rewrite the expressions in terms of exponential and find the derivatives

II. Find the derivative of the following problems

ANSWERS TO EXERCISE 2.9