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The product rule is a fundamental technique in calculus used for differentiating products of two functions. When you have a function that's a product of two simpler functions, say \[ f(x) = u(x) \cdot v(x) \], you can't simply take the derivative of each function separately and multiply them together. Instead, the product rule tells us how to differentiate:

• Identify the two parts of the product: \( u(x) \) and \( v(x) \).
• Differentiate each part individually, giving you \( u'(x) \) and \( v'(x) \).
• Apply the product rule formula: \( f'(x) = u'(x)v(x) + u(x)v'(x) \).

This means that for a product, not only must you take the derivative of each component, but also account for each function while multiplying by the derivative of the other. This ensures that any interaction between the two functions in the product is captured correctly. Forgetting this rule in scenarios like finding derivatives of products can lead to incorrect results.

The first derivative of a function, denoted by \( f'(x) \), represents the instantaneous rate of change of the function. It's like peeking into the function to see how it behaves and changes at a particular point. To make it clear:

• Think about a curve: the slope at any point on the curve is actually the first derivative.
• Calculating this involves taking each part of the original function and applying specific rules like the product rule.

In our problem with \( f(x) = x^2(x^4 - x) \), we use the product rule to calculate the first derivative:First, differentiate \( u(x) = x^2 \) to get \( u'(x) = 2x \), and \( v(x) = x^4-x \) to get \( v'(x) = 4x^3 - 1 \). Next, plug these into the product rule formula:\[ f'(x) = 2x(x^4 - x) + x^2(4x^3 - 1) \]Simplifying yields:\[ f'(x) = 6x^5 - 3x^2 \]This expression now represents how the original function changes and can be used to find slopes and movement along the function's curve.

The second derivative, denoted as \( f''(x) \), explores how the rate of change of a function itself changes. Think of it as the 'acceleration' of the original function. This step reveals how the slope of the function itself is increasing or decreasing. Here’s how it's approached:

• Once you have the first derivative \( f'(x) = 6x^5 - 3x^2 \), differentiate it again.
• Each term of the first derivative is treated separately, applying basic derivative rules.

For our example, differentiating each term yields:\[ f''(x) = 30x^4 - 6x \]This means the function’s slope is experiencing this new rate of change. Comparing this with the initial misquote of \( f''(x) \) shows the importance of correctly applying derivative rules. It uncovers mistakes like those in the problem, where they incorrectly attempted to find the second derivative by multiplying separate derivatives, leading to a false result. Understanding each derivative's role captures deeper features of the function's behavior.