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Magazine Chapter 3: Problem 17
In Exercises \(17-20,\) find \(d r / d \theta\) $$ r=4-\theta^{2} \sin \theta $$
Short Answer
Expert verified
\( \frac{dr}{d\theta} = -2\theta \sin \theta - \theta^2 \cos \theta \)
Step by step solution
01
Understand the Problem
We are given a function \( r(\theta) = 4 - \theta^2 \sin \theta \) and need to find the derivative \( \frac{dr}{d\theta} \).
02
Use the Product Rule
The function includes a product of \( \theta^2 \) and \( \sin \theta \). Recall the product rule for derivatives: \( (u \cdot v)' = u' \cdot v + u \cdot v' \). Here, \( u = \theta^2 \) and \( v = \sin \theta \).
03
Differentiate \( \theta^2 \sin \theta \)
Apply the product rule:- Derivative of \( u = \theta^2 \) is \( u' = 2\theta \).- Derivative of \( v = \sin \theta \) is \( v' = \cos \theta \).Thus, \( \frac{d}{d\theta}(\theta^2 \sin \theta) = 2\theta \cdot \sin \theta + \theta^2 \cdot \cos \theta \).
04
Differentiate the Constant Term
Differentiate the constant \( 4 \), which results in 0 because the derivative of a constant is always 0.
05
Combine Derivatives
Combine the derivatives from the previous steps. The derivative \( \frac{dr}{d\theta} \) is:\[ \frac{dr}{d\theta} = 0 - (2\theta \sin \theta + \theta^2 \cos \theta) \]Simplify to get:\[ \frac{dr}{d\theta} = -2\theta \sin \theta - \theta^2 \cos \theta \]
06
Simplify the Derivative
Our final expression for the derivative is \( -2\theta \sin \theta - \theta^2 \cos \theta \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Product Rule
In calculus, the **Product Rule** is a fundamental differentiation technique used when you have to differentiate a product of two functions. It's a crucial tool because many real-world problems involve interactions between multiple variables.
- The basic formula for the Product Rule is: \[ (u imes v)' = u' imes v + u imes v' \]where \( u \) and \( v \) are functions of \( x \), and \( u' \) and \( v' \) are their derivatives.
- For the function in our exercise, \( u = \theta^2 \) and \( v = \sin \theta \), we used this rule to find the derivative of their product.
- This technique allows us to handle more complex functions without simplifying them into a single term.
Trigonometric Derivatives
**Trigonometric Derivatives** are essential when dealing with functions that include trigonometric components. These derivatives help us understand how functions like \( \sin \theta \), \( \cos \theta \), and others change.
- For instance, from basic differentiation rules, we know that the derivative of \( \sin \theta \) is \( \cos \theta \).
- This piece of information is vital when applying the Product Rule to the given function \( r = 4 - \theta^2 \sin \theta \).
- By using this derivative appropriately, we identified the change in the sine component of the original function.
Differentiation Techniques
In the realm of calculus, different **Differentiation Techniques** provide the means to tackle a variety of function types. Recognizing which method applies to which function is crucial.
- The Product Rule, as discussed, is just one technique among many.
- Other methods include the Chain Rule, Quotient Rule, and basic power rule, each serving different function types.
- Choosing the correct method simplifies the process significantly, as seen in the step-by-step solution of the exercise.
Implicit Differentiation
Though not directly used in this exercise, **Implicit Differentiation** is a powerful technique when dealing with equations not easily solved for a single variable.
- It comes in handy when differentiating equations where variables are intermixed and cannot be isolated easily.
- Implicit Differentiation allows us to find derivatives without explicitly solving for one variable in terms of another.
- This method involves differentiating both sides of an equation simultaneously, often keeping terms related to different variables differentiated separately.
