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The chain rule is a fundamental technique in calculus used to differentiate composite functions. When a function is composed of two or more functions, the chain rule allows us to differentiate it efficiently. For example, if we have a function defined as

• Outer function: a function of another variable, like \[ u^2 \]
• Inner function: a function inside the outer, like \[ u = 3 - 2x \]

The chain rule states that the derivative of the composite function is the derivative of the outer function times the derivative of the inner function:
\[ \frac{dy}{dx} = \frac{du}{dx} \times \frac{dy}{du} \]
In the original exercise, this is applied as follows: ```latex\frac{dy}{dx} = 2(3 - 2x)^1 \cdot \frac{d}{dx}(3 - 2x)``` By substituting the derivative of the inner function, we complete the differentiation using the chain rule, making it a powerful tool for solving complex derivatives.

Derivatives represent the rate of change of a function with respect to one of its variables. It is one of the core concepts of calculus, providing insight into how a function behaves. The notation \( \frac{dy}{dx} \) indicates the derivative of \( y \) with respect to \( x \). To understand derivatives, consider two important steps:
- Find the rate of change of each function inside a composite or expanded function.
- Apply rules like the chain rule to simplify and consolidate the results.
In the example, after applying the chain rule to our function \( y = (3 - 2x)^2 \), we find the derivative of the inner function as \( -2 \) and multiply it with the derivative of the outer part, leading us towards the final step where \( \frac{dy}{dx} = 8x - 12 \).
Understanding derivatives helps in predicting functionality and behavior of variable changes over time.

Function expansion involves multiplying out all terms to rewrite a function into a format that is often more straightforward to differentiate.
Let us multiply given (3-2x)\( ^{2} \) in terms of its components:

• First term squared: \( 9 \)
• Middle terms and combining like terms: \( -6x - 6x = -12x \)
• Last term squared: \( 4x^2 \)

This results in expanding the expression as: ```latexy = 4x^2 - 12x + 9```Once expanded, differentiating the function becomes a direct process of applying basic derivative rules to each term. This alternative method of expansion is useful for checking accuracy of results post differentiation with more complex rules.

Working through problems systematically is crucial for mastering differentiation techniques. Here's how one would approach the original exercise:
1. **Apply the Chain Rule**: Identify the outer and inner functions in the composite and differentiate each.2. **Differentiate the Inner Function**: Solve for the derivative of the inner function; in our problem, it's a simple linear function.3. **Combine Results**: Multiply the derivatives from the chain rule application to get the initial simplified form.4. **Simplify if Needed**: Reduce the expression; often simplifying will give a clearer version of the derivative like we got \( 8x - 12 \).5. **Function Expansion**: Expand the original function and differentiate each term using basic rules.6. **Verify Results**: Make sure both methods agree to ensure accuracy. Following these steps will consistently help in understanding the process, enabling easier application to more complicated functions in the future.