Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-20, find \(D_{x} y\). $$ y=\sin ^{4}\left(3 x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative is \( 24x \sin^{3}(3x^{2}) \cos(3x^{2}) \).

Step 1: Understand the Problem

We need to find the derivative of the function \( y = \sin^{4}(3x^{2}) \) with respect to \( x \). This involves using both the chain rule and the power rule for derivatives.

Step 2: Apply the Power Rule

First, recognize that \( y = [\sin(3x^{2})]^{4} \) is an outer function raised to the power of 4. Using the power rule for derivatives, we have \( D_{x} u^{n} = n \cdot u^{n-1} \cdot u' \). Here \( n = 4 \) and \( u = \sin(3x^{2}) \). Therefore, the derivative of the outer function is \( 4 \cdot [\sin(3x^{2})]^{3} \cdot D_{x}(\sin(3x^{2})) \).

Step 3: Apply the Chain Rule

Next, we find the derivative of the inner function, \( \sin(3x^{2}) \), which requires the chain rule. The derivative of \( \sin(v) \) with respect to \( v \) is \( \cos(v) \). Setting \( v = 3x^{2} \), the derivative is \( \cos(3x^{2}) \cdot D_{x}(3x^{2}) \).

Step 4: Differentiate the Inner Function

The inner function is \( v = 3x^{2} \). Differentiate \( 3x^{2} \) to get \( 6x \).

Step 5: Combine Derivatives

Substitute back to get the derivative of the inner function: \( \cos(3x^{2}) \cdot 6x \). Now, substitute this into your result from the power rule application in Step 2. The derivative becomes \( 4 \cdot [\sin(3x^{2})]^{3} \cdot \cos(3x^{2}) \cdot 6x \).

Step 6: Simplify the Expression

Simplify the expression to get the final derivative: \( 24x \cdot \sin^{3}(3x^{2}) \cdot \cos(3x^{2}) \).

Key Concepts

Chain Rule Application

The chain rule is a fundamental technique for finding derivatives of composite functions—functions that are made up of one function inside another. When dealing with a function like \[ y = ext{sin}^4(3x^2) \]we notice it has an outer function: \( u^4 \), where \( u = ext{sin}(3x^2) \).
To apply the chain rule, you first take the derivative of the outer function (the power rule here, which we will cover later), and then multiply it by the derivative of the inner function. The objective is to "chain" or link these operations together to handle the complexity of nested functions.
In this example, identifying the outer and inner functions correctly is crucial for applying the chain rule successfully. Once identified, the chain rule helps in differentiating \( ext{sin}(3x^2) \), where the multiplication of derivatives provides the solution you need for solving the problem.

Power Rule for Derivatives

The power rule is an essential method used to find the derivative of any term that is raised to a power. If you have a function like \[ y = u^n \],the power rule states that \[ D_x(u^n) = n imes u^{n-1} imes D_x(u) \].Here, the power rule helps us differentiate the outer function. For the problem's function \( [ ext{sin}(3x^2)]^4 \), you set \( n = 4 \), and \( u = ext{sin}(3x^2) \).
Applying the power rule gives:

• First, multiply by the exponent: \( 4 imes ext{sin}^3(3x^2) \).
• Then, multiply by the derivative of the inner function, \( D_x( ext{sin}(3x^2)) \), via the chain rule.

This powerful tool simplifies finding derivatives by reducing steps otherwise needed if approached manually. Using this rule, you can approach complex expressions with nested terms more systematically and accurately.

Trigonometric Derivatives

Trigonometric derivatives are derivatives of functions involving trigonometric components such as sine, cosine, and others. For this exercise, \( ext{sin}(v) \) is a crucial part of the function \( ext{sin}^4(3x^2) \).
The derivative of \( ext{sin}(v) \) is \( ext{cos}(v) \). This allows us to differentiate the sine part when required. When combined with the inner derivative \( D_x(3x^2) \),you effectively tackle the trigonometric angle derivatives.
Understanding how to differentiate these trigonometric functions is critical because these functions often appear in calculus problems involving periodic or oscillatory motion. Distinguish between the basic derivatives:

• \( D_x( ext{sin}(x)) = ext{cos}(x) \)
• \( D_x( ext{cos}(x)) = - ext{sin}(x) \)

These rules make solving calculus problems with trigonometric parts more straightforward and allow you to predict changes in such functions more easily.

Differentiation Techniques

Differentiation techniques are methods used in calculus to determine the rate at which one quantity changes with respect to another. For the given function\( y = ext{sin}^4(3x^2) \),two main differentiation techniques were featured: the chain rule and the power rule.
Additionally, proper handling of trigonometric derivatives completes the necessary toolkit to effectively solve the derivative of a composite trigonometric function that incorporates raised powers.
These techniques are fundamental in calculus as they enable you to tackle very complex functions by breaking them down into simpler, more manageable parts. Practicing these techniques across various problem sets builds intuition for recognizing how to deploy each rule strategically:

• Breaking functions into smaller parts (chain rule application).
• Simplifying raised powered terms (power rule for derivatives).
• Mastering trigonometric function changes (trigonometric derivatives).

By mastering these methods, you can solve derivative problems with greater confidence and accuracy across diverse disciplines in calculus.