Problem 23 - University Calculus: Early Transcendentals

Question

Find \(d r / d \theta\). $$r=4-\theta^{2} \sin \theta$$

Step-by-Step Solution

Answer

\(\frac{dr}{d\theta} = -2\theta \sin \theta - \theta^2 \cos \theta\)

Step 1: Understand the Problem

We need to find the derivative of the given function \(r = 4 - \theta^2 \sin \theta\) with respect to the variable \(\theta\). This involves applying differentiation rules such as the product rule and chain rule.

Step 2: Differentiate Constant Term

Differentiate the constant term '4'. The derivative of a constant is zero. So, \(\frac{d}{d\theta}(4) = 0\).

Step 3: Apply Product Rule to \(\theta^2 \sin \theta\)

We need to differentiate \(\theta^2 \sin \theta\). Use the product rule: \(\frac{d}{d\theta}(uv) = u'v + uv'\), where \(u = \theta^2\) and \(v = \sin \theta\).

Step 4: Differentiate \(u = \theta^2\)

Calculate the derivative of \(u = \theta^2\). Using power rule, \(u' = 2\theta\).

Step 5: Differentiate \(v = \sin \theta\)

Calculate the derivative of \(v = \sin \theta\). The derivative of \(\sin \theta\) is \(\cos \theta\), leading to \(v' = \cos \theta\).

Step 6: Apply Product Rule

Substitute the derivatives back into the product rule formula: \(\frac{d}{d\theta}(\theta^2 \sin \theta) = (2\theta)(\sin \theta) + (\theta^2)(\cos \theta)\).

Step 7: Combine Results

Combine the results from the derivative of the constant and the differentiated product. Thus, \(\frac{dr}{d\theta} = 0 - [(2\theta)(\sin \theta) + (\theta^2)(\cos \theta)]\).

Step 8: Simplify the Expression

Simplify the final expression: \(\frac{dr}{d\theta} = -2\theta \sin \theta - \theta^2 \cos \theta\).

Key Concepts

Product Rule

The product rule is essential when differentiating functions that are multiplied together. In mathematical terms, if you have two functions, say \( u \) and \( v \), the derivative of their product is given by:

• \( \frac{d}{d\theta}(uv) = u'v + uv' \)

This formula tells us that you differentiate each function separately and then combine them. This will ensure that both parts of the product are accounted for in the differentiation.

For example, if \( u = \theta^2 \) and \( v = \sin \theta \), then first find \( u' = 2\theta \) using the power rule. Similarly, for \( v' \), differentiate \( \sin \theta \) to get \( \cos \theta \).

By applying the product rule, the differentiated expression of \( \theta^2 \sin \theta \) becomes:

• \( (2\theta)(\sin \theta) + (\theta^2)(\cos \theta) \)

This combined expression correctly captures the changes in both \( \theta^2 \) and \( \sin \theta \) as \( \theta \) varies.

Chain Rule

The chain rule is another differentiation tool that helps find the derivative of composite functions. Suppose you have a function composed of another function, such as \( f(g(\theta)) \). The chain rule states:

• \( \frac{d}{d\theta} f(g(\theta)) = f'(g(\theta)) \cdot g'(\theta) \)

This means you first differentiate the outer function and then multiply by the derivative of the inner function.

In the context of the given problem, if \( r = 4 - \theta^2 \sin \theta \), note that there isn't a straightforward use of the chain rule as it is directly a multiplication (handled by the product rule). However, understanding the chain rule is crucial for more complex scenarios where a function is nested.

It helps to visualize the process as peeling layers of an onion—differentiating from the outside in, addressing each layer separately.

Trigonometric Functions

Differentiating trigonometric functions is a common requirement in calculus. Two fundamental derivatives to remember are:

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

These derivatives are crucial when dealing with trigonometric expressions. In our exercise, we had \( r = 4 - \theta^2 \sin \theta \), and we needed to find \( \frac{d}{d\theta} \sin \theta \), which is \( \cos \theta \).

Trigonometric functions often appear in physics and engineering problems, where they describe periodic phenomena like waves. Knowing how to differentiate these functions is essential for analyzing such problems and understanding how the functions behave as their input changes.

So, whenever trigonometric functions are present, rely on these core derivative rules to guide you in the differentiation process.