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The Product Rule is a fundamental tool in calculus used for differentiating products of two functions. When finding the derivative of the product of two functions, say \( f(x) \) and \( g(x) \), the rule states that the derivative is equal to the derivative of the first function times the second function plus the first function times the derivative of the second function. Mathematically, this can be expressed as:

• \( (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \)

The Product Rule allows one to deconstruct the problem of differentiating complex products into two simpler tasks. For example, in the exercise given, to find the derivative of \( f(x)g(x) \) at \( x = 1 \), values are taken from the provided table and plugged into the product rule formula. This shows how the rule enables an effective and straightforward computation of derivatives for combinations of functions.

Function Tables are a method for organizing values and their corresponding derivatives to aid in computations, such as those required in finding derivatives using differentiation rules. In the exercise, the table provided values of \( f(x) \), \( f'(x) \), \( g(x) \), and \( g'(x) \) at specific points \( x = 1, 2, 3, 4 \). This setup makes it easier for someone who is solving the exercise to quickly locate and use the necessary values to apply differentiation rules like the Product Rule.

• In this instance, the table immediately highlights \( f(1) = 5 \), \( f'(1) = 3 \), \( g(1) = 4 \), and \( g'(1) = 2 \).

These tables are highly useful for quick reference and ensuring accurate computations, especially in exercises that require multiple values at specific points.

Mathematical Notation is crucial to clearly communicate mathematical ideas or principles, such as describing derivatives and the application of rules like the Product Rule. Notation like \( f'(x) \) is used to denote the derivative of a function \( f(x) \), which can shelter from confusion when analyzing or solving problems. In the context of the exercise, Mathematical Notation represents functions and their derivatives, aiding in straightforward identification and utilization in calculations:

• \( f'(1) \): Represents the derivative of \( f(x) \) evaluated at \( x = 1 \).
• \( \frac{d}{dx}(f(x)g(x))|_{x=1} \): Clearly denotes the derivative of the product of \( f(x) \) and \( g(x) \) at \( x = 1 \).

Such concise notations are essential for conveying complex procedures in a manageable form, making it easier for anyone to follow along with the mathematical process involved.