# Hyperbolic Functions

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In these lessons, we will look at Hyperbolic Functions, Hyperbolic Identities, Derivatives of Hyperbolic Functions and Derivatives of Inverse Hyperbolic Functions and how to evaluate them. We will look at the graphs of some Hyperbolic Functions and the proofs of some of the Hyperbolic Identities.

The following tables give the Definition of the Hyperbolic Function, Hyperbolic Identities, Derivatives of Hyperbolic Functions and Derivatives of Inverse Hyperbolic Functions. Scroll down the page for more examples and solutions.

Example:

Differentiate

Solution:

Using the table above and the Chain Rule.

Example:

Find the derivative of

Solution:

Using the above table and the Chain Rule

### Hyperbolic Functions and their Derivatives

Hyperbolic Functions - The Basics
This video gives the definitions of the hyperbolic functions,
a rough graph of three of the hyperbolic functions: y = sinh x, y = cosh x, y = tanh x
evaluate a few of the functions at different values: sinh(0), cosh(0), tanh(1)
and justify a couple of identities: cosh x + sinh x = ex, sinh(2x) = 2sinh x cosh x, sinh(2) = 2sinh x cosh x. Introduction to Hyperbolic Functions
This video provides a basic overview of hyperbolic function. The lesson defines the hyperbolic functions, shows the graphs of the hyperbolic functions, and gives the properties of hyperbolic functions.

Hyperbolic functions are exponential functions that share similar properties to trigonometric functions.

Hyperbolic Functions Properties
The point (cos(t), sin(t)) is on the unit circle x2 + y2 = 1.
The point (cosh(t), sinh(t)) is on the hyperbola x2 - y2 = 1.

Graphs of hyperbolic functions: f(x) = sinh(x), f(x) = csch(x), f(x) = cosh(x), f(x) = sech(x), f(x) = tanh(x), f(x) = coth(x).

Hyperbolic Functions in real life: Catenary
The catenary is the curve formed by a hanging cable or chain under its own weight when supported only at its ends. Hyperbolic Functions - Derivatives
This video shows the formulas for the derivatives of the hyperbolic functions and finds a few derivatives.
Examples: Find the derivative
f(x) = tanh(4x)
f(x) = sinh x tanh x

Inverse Hyperbolic Functions - Derivatives
This video gives the formulas for the derivatives on the inverse hyperbolic functions and does 3 examples of finding derivatives.
Example: Find the derivatives:
y = x2sinh-1(2x)
$$y = {\tanh ^{ - 1}}\sqrt x$$
$$y = \sec {h^{ - 1}}\sqrt {1 - x}$$

### Proof of Hyperbolic Identities

This video shows a proof of one of the properties of hyperbolic functions.
Prove a Property of Hyperbolic Functions: cosh2x - sinh2x = 1
Prove a Property of Hyperbolic Functions: (tanh(x))2 + (sech(x))2 = 1 Prove a Property of Hyperbolic Functions: sinh(x+y) = sinh(x)cosh(y)+cosh(x)sinh(y) Prove a Property of Hyperbolic Functions: (sinh(x))2 = (-1+cosh(2x))/2

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