Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the indicated derivative. $$ \frac{d}{d \theta}\left(\sin ^{3} \theta\right) $$

Step-by-Step Solution

Answer

The derivative is \( 3 \sin^2 \theta \cos \theta \).

Step 1: Identify the Outer Function

The expression can be rewritten as \( \left( \sin \theta \right)^3 \). Recognize that the outer function is a power function, specifically \( u^3 \), where \( u = \sin \theta \).

Step 2: Apply the Chain Rule

Use the chain rule to differentiate \( u^3 \) with respect to \( \theta \). The chain rule states that \( \frac{d}{d\theta} f(g(\theta)) = f'(g(\theta)) \times g'(\theta) \). Here, \( f(u) = u^3 \) and \( g(\theta) = \sin \theta \).

Step 3: Differentiate the Outer Function

Differentiate \( u^3 \) with respect to \( u \):\[\frac{d}{du} u^3 = 3u^2\]

Step 4: Differentiate the Inner Function

Differentiate \( \sin \theta \) with respect to \( \theta \):\[\frac{d}{d\theta} \sin \theta = \cos \theta\]

Step 5: Apply the Derivatives

Multiply the derivative of the outer function by the derivative of the inner function:\[3(\sin \theta)^2 \times \cos \theta\]

Step 6: Final Expression

The derivative of \( \sin^3 \theta \) with respect to \( \theta \) is:\[3 \sin^2 \theta \cos \theta\]

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus that helps compute the derivative of composite functions. Composite functions are functions nested within each other, like our example, \(\sin^3 \theta\), which is \((\sin \theta)^3\). When dealing with such functions, the chain rule allows us to differentiate by working from the inside out.

The chain rule formula is \(\frac{d}{d\theta} f(g(\theta)) = f'(g(\theta)) \times g'(\theta)\). Simply put, it tells us to take the derivative of the outer function while keeping the inner function unchanged, and then multiply that result by the derivative of the inner function. In our exercise, the outer function is the cube of the inner function \(u^3\), where \(u = \sin \theta\). This step-by-step process makes the task of finding the derivative more manageable.

Understanding and correctly applying the chain rule is crucial as it frequently appears in calculus problems involving composite functions.

Differentiation Techniques

Differentiation is the process of finding the derivative of a function, which can be thought of as measuring how a function's output changes as its input changes. In calculus, several techniques aid in simplifying this process.

- **Power Rule**: This is used when differentiating expressions like \(x^n\). The derivative is found by bringing down the power and reducing it by one: \(\frac{d}{dx} x^n = nx^{n-1}\). The power rule was used in Step 3 of the solution where we had \(u^3\) and found \(3u^2\).

- **Product Rule and Quotient Rule**: Though not directly used in this exercise, these rules are essential for dealing with products or quotients of functions.

By combining these techniques as needed – such as using the chain rule along with the power rule – complex differentiation problems become solvable.

Trigonometric Functions Differentiation

Differentiating trigonometric functions is a critical aspect of calculus, especially in fields involving periodic phenomena like waves or cycles. Common trigonometric functions include \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\), each with well-known derivatives.

- The derivative of \(\sin \theta\) is \(\cos \theta\). This was highlighted in Step 4 of the given solution.

- The derivative of \(\cos \theta\) is \(-\sin \theta\).
- The derivative of \(\tan \theta\) is \sec^2 \theta\.

Knowing these derivatives by heart allows for quick calculus work with trigonometric functions. Additionally, when we use these derivatives with rules like the chain rule, we can handle even more complex functions that incorporate trigonometry, as seen in the exercise. By methodically applying differentiation rules, the process becomes routine.