Problem 64 - Calculus - AP Edition

Question

True-False Determine whether the statement is true or false. Explain your answer. If \(y=\sin ^{3}\left(3 x^{3}\right),\) 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

True, the given derivative expression is correct.

Step 1: Understanding the Problem

We need to find the derivative \( \frac{dy}{dx} \) of the function \( y = \sin^3(3x^3) \), and verify if it matches with \( 27x^2 \sin^2(3x^3) \cos(3x^3) \).

Step 2: Identify the Chain Rule Usage

The function \( y = \sin^3(3x^3) \) indicates a composition of functions, where the outer function is \( u^3 \) and \( u = \sin(3x^3) \). We will apply the chain rule to differentiate.

Step 3: Differentiate using the Power Rule

Let \( u = \sin(3x^3) \). Then \( y = u^3 \). The derivative of \( y \) with respect to \( u \) is \( 3u^2 \).

Step 4: Differentiate the Inner Function

Now differentiate \( u = \sin(3x^3) \). The derivative is \( \frac{du}{dx} = \cos(3x^3) \cdot 9x^2 \), using the chain rule for \( \sin(v) \) where \( v = 3x^3 \).

Step 5: Apply Chain Rule to Find \( \frac{dy}{dx} \)

Using the chain rule, \( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = 3u^2 \cdot (\cos(3x^3) \cdot 9x^2) \). Substitute \( u = \sin(3x^3) \) to get \( \frac{dy}{dx} = 27x^2 \sin^2(3x^3) \cos(3x^3) \).

Step 6: Conclusion

After applying all calculation steps, the derived expression \( 27x^2 \sin^2(3x^3) \cos(3x^3) \) matches the initially provided derivative. Hence, the statement is true.

Key Concepts

Derivative

In calculus, a derivative represents how a function changes as its input changes. The derivative of a function at a point gives the slope of the tangent line to the function at that point. This concept is central to understanding motion, growth rates, and other dynamic processes.
When taking the derivative, we look at how a small change in the input leads to a change in the output.

• For the function \(y = \sin^3(3x^3)\), finding its derivative \(\frac{dy}{dx}\) involves understanding how the output \(y\) changes as \(x\) increases or decreases.
• The derivative can be computed using various rules like the product rule, quotient rule, and the chain rule, depending on the function's form.

In our case, we'll use the chain rule to determine how the composition of functions behaves when the input, \(3x^3\), is modified by \(x\). The derivative will tell us how the overall function changes accordingly.

Composition of functions

The composition of functions is a process of combining two functions to produce a new function. In simpler terms, you take the output of one function and use it as the input for another.

• In the given expression, \(y = \sin^3(3x^3)\), the composition is evident. The innermost function is \(3x^3\), followed by the \(\sin\) function, and finally, cubing the result.
• This multilayered approach makes it necessary to use the chain rule to find the derivative.

When dealing with such compositions, you differentiate each function layer step by step:1. Differentiate the innermost function \(3x^3\), giving \(\frac{d}{dx}(3x^3) = 9x^2\).2. Treat the \(\sin\) function of this derivative as the next layer, leading to another differentiation.Composing functions thus allows complex structures, like polynomials inside trig functions, to be systematically handled.

Power Rule

The power rule is a simple yet powerful tool in calculus for finding derivatives of expressions of the form \(u^n\), where \(u\) is a function of \(x\). The rule states that if \(y = x^n\), then the derivative \(\frac{dy}{dx} = nx^{n-1}\).

• In our function, the power rule is applied to the outermost part of the expression: \(\sin^3(3x^3)\), where the exponent on \(\sin\) is 3.
• This means taking the derivative of something to the power of 3 leads to multiplying by the exponent and reducing it by one: \(3 \cdot u^2\), where \(u = \sin(3x^3)\).

The power rule simplifies the differentiation of polynomial functions by reducing the power by one and multiplying by the original power. This method is critical when integrating with the chain rule to systematically derive the expression. Combining the power rule with other rules like the chain rule allows for managing more complex functions seamlessly.