Derivative Of Sinh And Cosh

Understanding the Derivatives of Sinh x and Cosh x: A Comprehensive Guide

Hyperbolic functions, like their trigonometric counterparts, play a crucial role in various branches of mathematics, physics, and engineering. Understanding their derivatives is fundamental to solving many problems involving calculus and differential equations. This article provides a comprehensive guide to deriving and understanding the derivatives of the hyperbolic sine (sinh x) and hyperbolic cosine (cosh x) functions, exploring their properties and applications. We'll delve into the underlying definitions and use both the limit definition of the derivative and the power series expansions to solidify our understanding.

Introduction to Hyperbolic Functions

Before we dive into the derivatives, let's briefly review the definitions of sinh x and cosh x. These functions are defined using exponential functions:

• sinh x = (e^x - e^-x)/2
• cosh x = (e^x + e^-x)/2

Notice the striking similarity to the Euler's formulas relating trigonometric functions to complex exponentials. However, hyperbolic functions are real-valued functions with distinct properties and applications, often appearing in problems involving catenaries, hanging cables, and certain differential equations.

Deriving the Derivative of sinh x using the Limit Definition

The derivative of a function f(x) is defined as the limit of the difference quotient as the change in x approaches zero:

f'(x) = lim (h→0) [(f(x + h) - f(x))/h]

Let's apply this definition to sinh x:

1. Substitute: Replace f(x) with sinh x = (e^x - e^-x)/2 in the limit definition.

2. Expand: Substitute (x + h) into the sinh function:

   sinh(x + h) = (e^(x+h) - e^-(x+h))/2 = (e^xe^h - e^-xe^-h)/2

3. Difference Quotient: Form the difference quotient:

   [(sinh(x + h) - sinh(x))/h] = [(e^xe^h - e^-xe^-h)/2 - (e^x - e^-x)/2]/h

4. Simplify: Combine the terms and factor out common terms:

   = [e^x(e^h - 1) + e^-x(1 - e^-h)] / (2h)

5. Apply the Limit: As h approaches 0, we can use the known limits:

   lim (h→0) [(e^h - 1)/h] = 1 and lim (h→0) [(1 - e^-h)/h] = 1

6. Final Result: Applying these limits, we get:

   d(sinh x)/dx = lim (h→0) [(e^x(e^h - 1) + e^-x(1 - e^-h)) / (2h)] = e^x/2 + e^-x/2 = cosh x

Therefore, the derivative of sinh x is cosh x.

Deriving the Derivative of cosh x using the Limit Definition

We follow a similar process for cosh x:

1. Substitute: Replace f(x) with cosh x = (e^x + e^-x)/2 in the limit definition.

2. Expand: Substitute (x + h) into the cosh function:

   cosh(x + h) = (e^(x+h) + e^-(x+h))/2 = (e^xe^h + e^-xe^-h)/2

3. Difference Quotient: Form the difference quotient:

   [(cosh(x + h) - cosh(x))/h] = [(e^xe^h + e^-xe^-h)/2 - (e^x + e^-x)/2]/h

4. Simplify: Combine the terms and factor out common terms:

   = [e^x(e^h - 1) - e^-x(e^-h - 1)] / (2h)

5. Apply the Limit: As h approaches 0, we utilize the known limits from the previous derivation.

6. Final Result: Applying the limits, we obtain:

   d(cosh x)/dx = lim (h→0) [(e^x(e^h - 1) - e^-x(e^-h - 1)) / (2h)] = e^x/2 - e^-x/2 = sinh x

Therefore, the derivative of cosh x is sinh x.

Deriving the Derivatives using Power Series Expansions

Another elegant approach involves using the power series expansions of sinh x and cosh x:

• sinh x = x + x³/3! + x⁵/5! + x⁷/7! + ...
• cosh x = 1 + x²/2! + x⁴/4! + x⁶/6! + ...

Differentiating these series term by term (a valid operation within their radii of convergence), we obtain:

• d(sinh x)/dx = 1 + x²/2! + x⁴/4! + x⁶/6! + ... = cosh x
• d(cosh x)/dx = x + x³/3! + x⁵/5! + x⁷/7! + ... = sinh x

This method elegantly confirms the derivatives we found using the limit definition. The power series approach highlights the inherent relationship between sinh x, cosh x, and their derivatives.

Higher-Order Derivatives

The cyclical nature of the derivatives of sinh x and cosh x is noteworthy. Let's examine the higher-order derivatives:

• First derivative of sinh x: cosh x
• Second derivative of sinh x: sinh x
• Third derivative of sinh x: cosh x
• And so on...

Similarly:

• First derivative of cosh x: sinh x
• Second derivative of cosh x: cosh x
• Third derivative of cosh x: sinh x
• And so on...

This pattern continues indefinitely, demonstrating a periodic nature in their higher-order derivatives.

Applications of Derivatives of Hyperbolic Functions

The derivatives of sinh x and cosh x are essential in several areas:

• Differential Equations: They appear frequently in solutions to second-order linear homogeneous differential equations with constant coefficients.
• Physics: They are used in describing the shape of a hanging cable (catenary), the motion of a damped harmonic oscillator, and other physical phenomena.
• Engineering: In structural engineering, they are used in analyzing the stress and strain in structures under tension.
• Calculus: They provide valuable examples in illustrating differentiation techniques and applications of calculus.

Frequently Asked Questions (FAQ)

Q: What is the difference between trigonometric and hyperbolic functions?

A: While both types of functions use similar names (sine, cosine, etc.), their definitions and properties differ significantly. Trigonometric functions are based on the unit circle, while hyperbolic functions are defined using exponential functions. Their graphs also exhibit distinct characteristics.

Q: Are there other hyperbolic functions besides sinh x and cosh x?

A: Yes, just as there are other trigonometric functions (tangent, cotangent, secant, cosecant), there are corresponding hyperbolic functions: tanh x, coth x, sech x, and csch x, each with its own derivative.

Q: How do I find the derivatives of the other hyperbolic functions (tanh x, coth x, etc.)?

A: You can find their derivatives using the quotient rule and the derivatives of sinh x and cosh x. For example, tanh x = sinh x / cosh x, so its derivative can be found using the quotient rule.

Q: What are some common mistakes to avoid when working with hyperbolic functions?

A: A common mistake is to confuse hyperbolic functions with trigonometric functions. Remember their distinct definitions and properties. Another mistake is to incorrectly apply trigonometric identities to hyperbolic functions; hyperbolic identities exist but are different.

Q: Are there any online resources or tools to help with calculations involving hyperbolic functions?

A: Many online calculators and software packages (like Wolfram Alpha or mathematical software such as Maple or Mathematica) can handle calculations involving hyperbolic functions and their derivatives.

Conclusion

Understanding the derivatives of sinh x and cosh x is a fundamental step in mastering calculus and its applications. We've explored different methods for deriving these derivatives, highlighting their cyclical nature and their importance in various fields. By grasping these concepts, you will be better equipped to solve problems involving differential equations, physics, engineering, and advanced mathematical analysis. Remember that consistent practice and a thorough understanding of the underlying principles are key to mastering these functions and their applications. Don't hesitate to review the different approaches provided here and practice deriving the derivatives independently to solidify your understanding.