Problem 1 - Calculus and its applications

Question

Differentiate each function $$ y=(3-2 x)^{2} $$ Check by expanding and then differentiating.

Step-by-Step Solution

Answer

The derivative of the function is \( \frac{dy}{dx} = 8x - 12 \).

Step 1: Use the Chain Rule

To differentiate the function \( y = (3 - 2x)^2 \), we first apply the chain rule which states that \( \frac{dy}{dx} = 2(3 - 2x)^1 \cdot \frac{d}{dx}(3 - 2x) \).

Step 2: Differentiate the Inner Function

Differentiate the inner function \( 3 - 2x \) with respect to \( x \). The derivative is \( -2 \).

Step 3: Combine Chain Rule Results

Multiply the results from the chain rule: \( \frac{dy}{dx} = 2(3 - 2x)^1 \cdot (-2) = -4(3 - 2x) \).

Step 4: Simplify the Expression

Simplify \( -4(3 - 2x) \) to get \( -12 + 8x \). Thus, the derivative \( \frac{dy}{dx} = 8x - 12 \).

Step 5: Expand the Original Function

Expand \( y = (3 - 2x)^2 \) to \( y = (3 - 2x)(3 - 2x) = 9 - 6x - 6x + 4x^2 = 4x^2 - 12x + 9 \).

Step 6: Differentiate the Expanded Function

Differentiate \( y = 4x^2 - 12x + 9 \) term by term, resulting in \( \frac{dy}{dx} = 8x - 12 \).

Step 7: Verify Consistency of Results

Both the differentiation by the chain rule method and the expansion method give the same derivative: \( \frac{dy}{dx} = 8x - 12 \).

Key Concepts

Chain Rule

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.

Derivative

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

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.

Step-by-Step Solution

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.