Problem 13 - Thomas Calculus

Question

In Exercises \(13-24,\) find the derivative of \(y\) with respect to the appropriate variable. $$ y=6 \sinh \frac{x}{3} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = 2 \cosh \left( \frac{x}{3} \right) \).

Step 1: Understand the Hyperbolic Sine Function

The hyperbolic sine function is defined as \( \sinh u = \frac{e^u - e^{-u}}{2} \). The derivative of \( \sinh u \) with respect to \( u \) is \( \cosh u \).

Step 2: Identify the Inner Function

In the function \( y = 6 \sinh \frac{x}{3} \), the inner function is \( u = \frac{x}{3} \). We will need to differentiate this inner function as part of the chain rule.

Step 3: Differentiate the Inner Function

Differentiate \( u = \frac{x}{3} \) with respect to \( x \). The derivative is \( \frac{du}{dx} = \frac{1}{3} \).

Step 4: Differentiate the Outer Function

Apply the derivative of the hyperbolic sine function. We have \( y = 6 \sinh u \), and the derivative of \( \sinh u \) is \( \cosh u \). So, \( \frac{dy}{du} = 6 \cosh u \).

Step 5: Apply the Chain Rule

To find \( \frac{dy}{dx} \), apply the chain rule: \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \). Substitute the derivatives: \( \frac{dy}{dx} = 6 \cosh \left( \frac{x}{3} \right) \times \frac{1}{3} \).

Step 6: Simplify the Derivative

Simplify the expression to find the derivative of \( y \) with respect to \( x \). The derivative is \( \frac{dy}{dx} = 2 \cosh \left( \frac{x}{3} \right) \).

Key Concepts

Derivative

A derivative is a concept in differential calculus that represents the rate at which a function changes relative to its variable. It essentially gives us the slope of the function's tangent line at any given point. For example, in the function \( y = 6 \sinh \frac{x}{3} \), finding the derivative means determining how \( y \) changes as \( x \) changes.
Derivatives are powerful tools to understand the behavior of functions. They help in studying motion, growth, and change in various contexts. When you differentiate a function, you are looking for how small changes in input produce changes in the output.
To calculate the derivative, we often follow systematic steps, which may involve using rules like the product rule, quotient rule, or the chain rule. Understanding these rules is essential for computing derivatives effectively.

Chain Rule

The chain rule is a crucial concept in calculus used when differentiating composite functions—functions made up of other functions. When you have a function within a function, the chain rule helps you find the derivative.
For instance, consider the function \( y = 6 \sinh \left( \frac{x}{3} \right) \). Here, \( \sinh \) is the outer function and \( \frac{x}{3} \) is the inner function. The chain rule states that to find the derivative of a composite function \( y = f(g(x)) \), you first differentiate the outer function with respect to the inner function, and then multiply by the derivative of the inner function.

• Differentiating the outer function \( 6 \sinh u \) gives \( 6 \cosh u \).
• The inner function \( u = \frac{x}{3} \) has the derivative \( \frac{1}{3} \).

Thus, using the chain rule, the derivative \( \frac{dy}{dx} \) is calculated as \( \frac{dy}{du} \times \frac{du}{dx} = 6 \cosh \left( \frac{x}{3} \right) \times \frac{1}{3} = 2 \cosh \left( \frac{x}{3} \right) \).
This technique is invaluable whenever functions have layers, ensuring each part is differentiated correctly.

Hyperbolic Functions

Hyperbolic functions are analogues of trigonometric functions but are based on hyperbolas instead of circles. They are widely used in various scientific fields, particularly in hyperbolic geometry and calculus.
Some common hyperbolic functions include \( \sinh \, u \), \( \cosh \, u \), \( \tanh \, u \), and their inverses. In differential calculus, knowing the derivatives of these functions is essential. For instance, the derivative of \( \sinh u \) is \( \cosh u \), just as the derivative of \( \sin u \) in trigonometry is \( \cos u \).
The function used in our example, \( \sinh \frac{x}{3} \), behaves much like sine, but follows a hyperbolic curve. Understanding how to differentiate hyperbolic functions allows calculation of changes in scenarios that apply to real-world contexts, such as physics and engineering. Just like with trigonometric functions, hyperbolic functions come with their unique identities and characteristics that are crucial for mathematical modeling and analysis.
By gaining clarity on hyperbolic functions, students can better understand their behavior and applications.