Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\). $$ y=\sin ^{4}\left(3 x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative is \( D_x y = 24x (\sin(3x^2))^3 \cos(3x^2) \).

Step 1: Identify the Function Structure

The expression provided is a composite function. We have an outer function and an inner function: \(y = \sin^4(3x^2)\), which can be rewritten as \(y = (\sin(3x^2))^4\).

Step 2: Apply the Chain Rule

To differentiate the function, apply the chain rule. First, set \(u = \sin(3x^2)\) such that \(y = u^4\). Then, differentiate \(y\) with respect to \(u\), obtaining \(\frac{dy}{du} = 4u^3\).

Step 3: Differentiate Inner Function with Respect to x

Next, differentiate the inner function \(u = \sin(3x^2)\) with respect to \(x\). Use the chain rule: \(\frac{du}{dx} = \cos(3x^2) \cdot \frac{d}{dx}(3x^2)\). The derivative of \(3x^2\) is \(6x\), so \(\frac{du}{dx} = \cos(3x^2) \cdot 6x\).

Step 4: Combine Results Using the Chain Rule

Multiply the derivatives found in Step 2 and Step 3 to find \(\frac{dy}{dx}\):\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 4u^3 \cdot 6x \cdot \cos(3x^2)\].Substitute \(u = \sin(3x^2)\) back into this equation:\[\frac{dy}{dx} = 24x \cdot (\sin(3x^2))^3 \cdot \cos(3x^2)\].

Step 5: Write the Final Answer

The derivative of \(y = \sin^4(3x^2)\) with respect to \(x\) is:\[D_x y = 24x \cdot (\sin(3x^2))^3 \cdot \cos(3x^2)\]. That is the solution to the derivative question.

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of composite functions. Simply put, the chain rule allows us to differentiate complex expressions by breaking them into simpler parts.
This rule states that if you have a composite function, denoted as \( y = f(g(x)) \), the derivative \( \frac{dy}{dx} \) is found by taking the derivative of the outer function \( f \) while holding the inner function \( g \) fixed, then multiplying by the derivative of the inner function \( g \).
The chain rule formula looks like this:

• \( \frac{dy}{dx} = \frac{df}{dg} \times \frac{dg}{dx} \)

This is a powerful tool, especially when dealing with nested functions or when applying it multiple times in succession.
In the original exercise, we identified two separate functions interacting: the outer \( f(u) = u^4 \) and the inner \( g(x) = \sin(3x^2) \), thus allowing us to systematically differentiate each using the chain rule.

Composite Functions

A composite function combines two or more functions into a single expression. This occurs when one function is nested inside another. For instance, consider \( h(x) = f(g(x)) \). Here, \( g(x) \) is the inner function, and \( f \) is the outer function, as it operates on the result of \( g(x) \). Understanding and identifying composite functions is crucial for applying the chain rule.
A simple illustration might be temperature conversion: converting Celsius to Fahrenheit involves two functions—one adding 32 degrees and another multiplying by 9/5.
In our exercise, the function \( y = \sin^4(3x^2) \) consists of an outer component \( (\sin(u))^4 \) and an inner component \( u = 3x^2 \). Recognizing this structure makes differentiation more manageable and less prone to error.

Trigonometric Differentiation

Trigonometric differentiation involves finding the derivative of trigonometric functions, which are essential in many areas of math and science. The basic derivatives include:

• \(\frac{d}{dx}(\sin x) = \cos x\)
• \(\frac{d}{dx}(\cos x) = -\sin x\)
• \(\frac{d}{dx}(\tan x) = \sec^2 x\)

These derivatives form the building blocks for more complex expressions involving trigonometric functions.
During differentiation, you often use these basic derivatives, possibly multiple times, particularly when applying the chain rule or working with composite trigonometric functions.
In our example, the derivative of \( \sin(3x^2) \) is particularly interesting. You start by differentiating the inner function, \( 3x^2 \), which yields \( 6x \). Then apply the chain rule combining with the trigonometric derivative \( \cos(3x^2) \), resulting in the complex expression essential to our final solution: \( 24x \cdot (\sin(3x^2))^3 \cdot \cos(3x^2) \). Understanding these trigonometric principles is crucial in correctly solving such derivatives.