Problem 64 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. $$ \text { If } y=\sin ^{3}\left(3 x^{3}\right), \text { then } d y / d x=27 x^{2} \sin ^{2}\left(3 x^{3}\right) \cos \left(3 x^{3}\right) $$

Step-by-Step Solution

Answer

The statement is true; the calculated derivative matches the given expression.

Step 1: Identify the Function

The given function is \( y = \sin^3(3x^3) \). This means that \( y \) involves \( \sin^3(u) \) where \( u = 3x^3 \).

Step 2: Apply the Chain Rule

To find \( \frac{dy}{dx} \), recognize that it requires the chain rule for differentiation. First, differentiate the outer function \( y = u^3 \) with respect to \( u \), giving \( 3u^2 \cdot \frac{du}{dx} \) where \( u = \sin(3x^3) \).

Step 3: Differentiate the Sine Function

Now, differentiate \( u = \sin(3x^3) \) with respect to \( x \). This involves another chain rule: The derivative of \( \sin(t) \) with respect to \( t \) is \( \cos(t) \), multiplied by the derivative of \( t = 3x^3 \), giving \( 9x^2 \cos(3x^3) \).

Step 4: Combine the Derivatives

Combine the derivatives: \( \frac{dy}{du} = 3\sin^2(3x^3) \) and \( \frac{du}{dx} = 9x^2\cos(3x^3) \). Thus, \( \frac{dy}{dx} = 3\sin^2(3x^3) \cdot 9x^2\cos(3x^3) \).

Step 5: Simplify to Match Given Solution

Now, simplify \( \frac{dy}{dx} = 27x^2\sin^2(3x^3)\cos(3x^3) \). This matches the given expression for \( \frac{dy}{dx} \).

Step 6: Conclusion

Since the calculated derivative exactly matches the given expression, the statement that \( \frac{dy}{dx} = 27x^2\sin^2(3x^3)\cos(3x^3) \) is true.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, which involves finding the derivative of a function. The derivative measures how a function's output value changes concerning its input value. In simpler terms, it's the rate of change or slope of the function. For example, in our exercise, we have a composite function: \( y = \sin^3(3x^3) \). Finding \( \frac{dy}{dx} \), the derivative, requires understanding how changes in \( x \) affect \( y \).

The chain rule is particularly useful for differentiating composite functions, which are functions within other functions. It allows us to break down complex functions into manageable parts and differentiate each piece. If you imagine a function as a series of machines, the chain rule helps us determine how adjustments at the start affect the final outcome.

When you apply the chain rule, consider:

• Identifying and differentiating the outer function.
• Differentiating any inner functions within.
• Combining these derivatives to find the overall derivative of the composite function.

This method is essential for calculus problems where terms are nested within one another.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are pivotal in calculus and many applications across sciences and engineering. These functions relate angles in a right triangle to ratios of the triangle's sides, and they are crucial for studying wave patterns, oscillations, and much more.

In our exercise, the function \( y = \sin^3(3x^3) \) involves the sine function elevated to the third power and an argument of \( 3x^3 \). Differentiating such functions becomes complex due to their cyclical properties and requires careful attention through chain rule applications.

The key steps you should remember when differentiating trigonometric functions:

• The derivative of \( \sin(t) \) is \( \cos(t) \).
• Where the argument of the function (e.g., \( 3x^3 \)) requires its derivative, multiplying together gets you the chain rule output.
• Combine these derivatives to find the rate of change of the trigonometric expression concerning \( x \).

Mastering these concepts ensures you have a strong grasp on handling periodic functions like sine and cosine within calculus.

Simplification of Expressions

Simplification of expressions in calculus means reducing them to a more manageable form while preserving their value. In our problem, after applying the chain rule and finding the derivative, we obtained an expression: \( 3\sin^2(3x^3) \cdot 9x^2\cos(3x^3) \). Simplification makes it easier to read and also confirms agreement with initial problem statements.

During simplification, you can:

• Combine coefficients: In this exercise, multiplying \( 3 \) and \( 9 \) to get \( 27 \).
• Recognize common terms: Here, \( \sin^2(3x^3) \) and \( \cos(3x^3) \) are arranged neatly, reflecting the product of sine squared and cosine.
• Ensure that your final expression matches any given formats to verify solution correctness.

Effective simplification helps both in problem-solving and communicating your answer clearly to others. Always check your work meticulously to ensure accuracy after any simplifications.