Problem 49 - University Calculus: Early Transcendentals

Question

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\frac{\theta+5}{\theta \cos \theta}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{d\theta} = \frac{1}{\theta \cos \theta} - \frac{1}{\theta^2 \cos \theta} - \frac{(\theta + 5) \sin \theta}{\theta \cos^2 \theta} \).

Step 1: Rewriting with Logarithms

Given the function \( y = \frac{\theta + 5}{\theta \cos \theta} \), first take the natural logarithm of both sides to make the expression more manageable for differentiation. This will be \( \ln y = \ln\left(\frac{\theta + 5}{\theta \cos \theta}\right) \). Using properties of logarithms: \( \ln y = \ln(\theta + 5) - \ln(\theta) - \ln(\cos \theta) \).

Step 2: Differentiate Both Sides

Differentiate both sides with respect to \( \theta \). For \( \ln y \), the derivative is \( \frac{1}{y} \frac{dy}{d\theta} \). For the right side, differentiate each term separately: the derivative of \( \ln(\theta + 5) \) is \( \frac{1}{\theta + 5} \), the derivative of \( \ln(\theta) \) is \( \frac{1}{\theta} \), and the derivative of \( \ln(\cos \theta) \) is \( -\tan \theta \).

Step 3: Combine Derivatives

Combine the derivatives: \( \frac{1}{y} \frac{dy}{d\theta} = \frac{1}{\theta + 5} - \frac{1}{\theta} - \tan \theta \).

Step 4: Solve for \( \frac{dy}{d\theta} \)

Multiply both sides by \( y \) to solve for \( \frac{dy}{d\theta} \). Substitute \( y = \frac{\theta + 5}{\theta \cos \theta} \) back into the equation: \( \frac{dy}{d\theta} = \left( \frac{1}{\theta + 5} - \frac{1}{\theta} - \tan \theta \right) \frac{\theta + 5}{\theta \cos \theta} \).

Step 5: Simplify the Final Expression

Distribute and simplify the expression to get the final derivative in a simpler form: \( \frac{dy}{d\theta} = \frac{1}{\theta \cos \theta} - \frac{1}{\theta^2 \cos \theta} - \frac{(\theta + 5) \sin \theta}{\theta \cos^2 \theta} \).

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes. In simpler terms, the derivative of a function gives us the slope of the function at any given point. For our exercise, we are interested in finding the derivative of a function involving the variable \( \theta \). Using standard differentiation techniques can often be complex, especially with functions in fraction form or those with products. Logarithmic differentiation simplifies this process, allowing us to break down products and quotients more easily by using properties of logarithms. Once parts of the function are simplified log-wise, applying the derivative rules to each part becomes straightforward. This method is particularly useful in handling complex functions like \( y = \frac{\theta+5}{\theta \cos \theta} \), where direct application of standard rules would be cumbersome.

Role of Natural Logarithms

Natural logarithms, denoted as \( \ln \), are logarithms with a base of \( e \), where \( e \approx 2.71828 \). They play a crucial role in calculus when simplification of complex functions is required, especially for differentiating products and powers. When we apply the natural logarithm to both sides of an equation, it enables us to leverage logarithmic identities to break down expressions into simpler components. In the current exercise, applying the natural logarithm allows us to rewrite the quotient \( \frac{\theta+5}{\theta \cos \theta} \) as a difference of logarithmic functions: \( \ln(\theta+5) - \ln(\theta) - \ln(\cos \theta) \). This step converts our complex division problem into a straightforward sum and difference problem, making the differentiation process much more manageable.

Trigonometric Functions in Differentiation

Trigonometric functions, such as sine, cosine, and tangent, are essential tools in mathematics. Understanding their rules and behaviors is vital when dealing with functions involving angles like \( \theta \). In our problem, the presence of \( \cos \theta \) in the denominator adds complexity to the derivative process.When differentiating trigonometric functions, it helps to know their specific derivatives. For instance, the derivative of \( \cos \theta \) is \( -\sin \theta \), while the derivative of \( \sin \theta \) is \( \cos \theta \). These derivatives help us form the overall derivative of more complex structures by recognizing patterns and applying the chain and product rules effectively. In the exercise, identifying that \( -\tan \theta = - \frac{\sin \theta}{\cos \theta} \) helps simplify the expression, ensuring that the final differentiation is accurate and comprehensible. Through this approach, we achieve a derivative that is both simplified and applicable to real-world scenarios.