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More Calculus Lessons

Calculus Games

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.

### 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 = e^{x}, 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 x^{2} + y^{2} = 1.

The point (cosh(t), sinh(t)) is on the hyperbola x^{2} - y^{2} = 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 = x^{2}sinh^{-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: cosh^{2}x - sinh^{2}x = 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

More Calculus Lessons

Calculus Games

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

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 = e

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 x

The point (cosh(t), sinh(t)) is on the hyperbola x

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.

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

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 = x

\(y = {\tanh ^{ - 1}}\sqrt x \)

\(y = \sec {h^{ - 1}}\sqrt {1 - x} \)

Prove a Property of Hyperbolic Functions: cosh

Prove a Property of Hyperbolic Functions: (tanh(x))

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