Problem 18 - Thomas Calculus

Question

In Exercises \(17-20,\) find \(d r / d \theta\) $$ r=\theta \sin \theta+\cos \theta $$

Step-by-Step Solution

Answer

The derivative \( \frac{dr}{d\theta} \) is \( \theta \cos \theta \).

Step 1: Identify the Function

We are given the function \( r(\theta) = \theta \sin \theta + \cos \theta \). Our task is to find the derivative \( \frac{dr}{d\theta} \).

Step 2: Apply the Sum Rule for Derivatives

The function \( r(\theta) = \theta \sin \theta + \cos \theta \) is a sum of two functions. We need to apply the sum rule for derivatives which states that the derivative of a sum of functions is the sum of their derivatives. So, \( \frac{d}{d\theta}(\theta \sin \theta + \cos \theta) = \frac{d}{d\theta}(\theta \sin \theta) + \frac{d}{d\theta}(\cos \theta) \).

Step 3: Differentiate \(\theta \sin \theta\)

We use the product rule \( (uv)' = u'v + uv' \) to differentiate \( \theta \sin \theta \). Let \( u = \theta \) and \( v = \sin \theta \), so \( u' = 1 \) and \( v' = \cos \theta \). The derivative is \( \frac{d}{d\theta}(\theta \sin \theta) = 1 \cdot \sin \theta + \theta \cdot \cos \theta = \sin \theta + \theta \cos \theta \).

Step 4: Differentiate \(\cos \theta\)

The derivative of \( \cos \theta \) with respect to \( \theta \) is \( -\sin \theta \). Thus, \( \frac{d}{d\theta}(\cos \theta) = -\sin \theta \).

Step 5: Combine the Derivatives

Now, we combine the derivatives found in steps 3 and 4. So, the derivative \( \frac{dr}{d\theta} = (\sin \theta + \theta \cos \theta) + (-\sin \theta) \).

Step 6: Simplify the Expression

Combine the terms to simplify: \( \frac{dr}{d\theta} = \sin \theta + \theta \cos \theta - \sin \theta = \theta \cos \theta \).

Key Concepts

Sum Rule for Derivatives

When dealing with derivatives, the Sum Rule is a fundamental concept that simplifies complex differentiation tasks. It states that the derivative of a sum of two (or more) functions is simply the sum of their individual derivatives. This can be expressed as: \[\frac{d}{d\theta}(f(\theta) + g(\theta)) = \frac{d}{d\theta}(f(\theta)) + \frac{d}{d\theta}(g(\theta)).\] In practical terms, if you're faced with a function that is a sum, such as \( r(\theta) = \theta \sin \theta + \cos \theta \), you can differentiate each term individually and then add the results together.
This rule is particularly helpful because it breaks down potentially complicated expressions into simpler pieces. You handle each component separately, making the differentiation process more straightforward and less error-prone.

• Differentiate each term separately.
• Add the resulting derivatives together.

Using the Sum Rule saves time and effort by allowing you to focus individually on specific parts of a larger expression. For the example in this exercise, the rule helped us separate the process of differentiating \( \theta \sin \theta \) from differentiating \( \cos \theta \), simplifying our task.

Product Rule

The Product Rule is essential when you need to differentiate a product of two functions. Imagine you have functions \( u(\theta) \) and \( v(\theta) \). The Product Rule tells us that the derivative of the product \( uv \) is given by: \[(u \cdot v)' = u'v + uv'.\] This means you differentiate one function while keeping the other constant, and then switch roles.
In our task of differentiating \( \theta \sin \theta \), consider:

• \(u = \theta\)
• \(v = \sin \theta\)
• \(u' = 1\)
• \(v' = \cos \theta\)

Applying the Product Rule here, we find \( \frac{d}{d\theta}(\theta \sin \theta) = 1 \cdot \sin \theta + \theta \cdot \cos \theta = \sin \theta + \theta \cos \theta \).
This rule is crucial because it enables you to handle products of functions, which are common in calculus. Without it, differentiating products would be much more challenging.

Trigonometric Functions

Trigonometric functions are fundamental in calculus, especially when dealing with periodic processes or rotational motion. Here, we encounter functions like \( \sin \theta \) and \( \cos \theta \), which have specific derivatives.

• The derivative of \( \sin \theta \) is \( \cos \theta \).
• The derivative of \( \cos \theta \) is \( -\sin \theta \).

These derivatives play a critical role when applying differentiation rules in problems that involve trigonometric expressions.
Recognizing and memorizing the derivatives of sine and cosine can greatly speed up your math work. They show up frequently and understanding them fully can lead to a more efficient problem-solving process.

• Always write down the specific derivative relationships for reference when you start differentiating trigonometric functions.
• Use these derivatives to transform your trigonometric expressions into simplified forms during calculations.

In the given exercise, knowing the derivative of \( \cos \theta \) was critical to finding \( \frac{dr}{d\theta} \), allowing the expression \( \theta \cos \theta \) to emerge as the solution. Recognizing these derivatives ensures accuracy and precision in your work.