Problem 30 - Calculus Early Transcendentals

Question

Compute \(d y / d x\) for the following functions. \(y=x / \operatorname{csch} x\)

Step-by-Step Solution

Answer

Based on the given step by step solution, the derivative of the function \(y = \frac{x}{\operatorname{csch}x}\) is: \(\frac{dy}{dx} = \frac{1 + x \operatorname{coth}x}{(\sinh x)^2} \operatorname{csch}^2x\)

Step 1: Find the derivative of \(\operatorname{csch}x\)

Recall that \(\operatorname{csch}x = \frac{1}{\sinh x}\). First, we will find the derivative of \(\sinh x\). We know that \(\sinh x = \frac{e^x - e^{-x}}{2}\). Using the chain rule, we can find the derivative of \(\sinh x\): \(\frac{d}{dx}(\sinh x) = \frac{d}{dx}\left(\frac{e^x - e^{-x}}{2}\right)\) Apply the chain rule: \(\frac{d}{dx}(\sinh x) = \frac{1}{2}\left(e^x + e^{-x}\right) = \cosh x\) Now, we can find the derivative of \(\operatorname{csch}x\) using the chain rule and the fact that \(\frac{d}{dx}\left(\frac{1}{u}\right) = \frac{-u'}{u^2}\): \(\frac{d}{dx}(\operatorname{csch}x) = \frac{d}{dx} \left(\frac{1}{\sinh x}\right) = -\frac{\frac{d}{dx}(\sinh x)}{(\sinh x)^2} = -\frac{\cosh x}{(\sinh x)^2}\)

Step 2: Apply the quotient rule to find the derivative of the given function

Now, we are ready to use the quotient rule to find the derivative of the given function \(y=\frac{x}{\operatorname{csch}x}\). Recall the quotient rule states that: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{vu' - uv'}{v^2}\) Applying the quotient rule, we have: \(\frac{dy}{dx} = \frac{d}{dx}\left(\frac{x}{\operatorname{csch}x}\right) = \frac{\operatorname{csch}x \cdot \frac{d}{dx}(x) - x \cdot \frac{d}{dx}(\operatorname{csch}x)}{(\operatorname{csch}x)^2}\) Now substitute the derivative of \(\operatorname{csch}x\) from Step 1: \(\frac{dy}{dx} = \frac{\operatorname{csch}x \cdot 1 - x\left(-\frac{\cosh x}{(\sinh x)^2}\right)}{(\operatorname{csch}x)^2}\) Simplify: \(\frac{dy}{dx} = \frac{\frac{1}{\sinh{x}} + \frac{x\cosh{x}}{(\sinh{x})^2}}{(\operatorname{csch}x)^2} = \frac{1 + x \operatorname{coth}x}{(\sinh x)^2} \operatorname{csch}^2x\) Thus, the derivative of the given function is: \(\frac{dy}{dx} = \frac{1 + x \operatorname{coth}x}{(\sinh x)^2} \operatorname{csch}^2x\)

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of the traditional trigonometric functions but based on hyperbolas instead of circles. They frequently appear in calculus and related fields, particularly when dealing with exponential functions and certain area problems. Two key hyperbolic functions are the hyperbolic sine, \(\sinh x\), and the hyperbolic cosecant, \(\operatorname{csch} x\).

- The hyperbolic sine function is defined as \(\sinh x = \frac{e^x - e^{-x}}{2}\).- The hyperbolic cosecant, which is the reciprocal of the hyperbolic sine, is given by \(\operatorname{csch} x = \frac{1}{\sinh x}\).

Just like trigonometric identities, hyperbolic identities also exist and have similar properties useful in various calculus problems. Understanding these foundational equations allows us to manipulate and differentiate hyperbolic functions fields.

Quotient Rule

The quotient rule is a crucial tool in calculus to find the derivative of a function that is the division of two differentiable functions. It is essential when dealing with rational functions where one function is divided by another. The formula of the quotient rule is:

\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}\]

Here, \(u\) and \(v\) are two differentiable functions of \(x\), and \(u'\) and \(v'\) are their respective derivatives. It's important to remember that you multiply the derivative of the numerator by the original denominator, subtract the product of the numerator and the derivative of the denominator, and then divide everything by the square of the denominator.

• Useful for differentiating rational expressions.
• Keeps track of the order of derivatives crucial for correct solutions.

This rule ensures that differentiating fractions of functions is systematic and precise.

Chain Rule

The chain rule is used when dealing with composite functions, those built from two or more functions. It allows us to find the derivative of functions nested within one another. Suppose you have a function \(y = f(g(x))\), the derivative using the chain rule is:

\[\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\]

This rule takes the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. The chain rule is indispensable when differentiating certain complex expressions, such as hyperbolic functions derived from exponential functions, as we saw earlier.

• Apply to functions of functions for correct differentiation.
• Especially useful when one function is embedded inside another.

Understanding the chain rule is pivotal in unravelling nested functions in calculus.

Calculus

Calculus is a broad field of mathematics focused on change and motion. It is divided into two main branches: differential calculus and integral calculus. Differential calculus, which we focus on here, involves the concept of the derivative. The derivative measures how a function changes as its input changes.

A strong insight into calculus affords understanding of fundamental principles underpinning the dynamics of systems in science and engineering. Derivatives are used to:

• Calculate the rate of change.
• Find tangents to curves.
• Determine maximum and minimum values of functions.

These applications are vital across physics, biology, economics, and beyond, serving as the backbone of many theoretical and practical fields. Mastery of differentiation, including techniques like the quotient and chain rules, is essential to unlock the full power of calculus.