Integration by parts [math: ∫f′(x)g(x)dx=f(x)g(x)−∫f(x)g′(x)dx]\int f'(x) g(x) \, dx = f(x) g(x) - \int f(x) g'(x) \, dx is the partial integral counterpart to which of the following rules?

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The question asks: Integration by parts, expressed as ∫ f'(x) g(x) dx = f(x) g(x) − ∫ f(x) g'(x) dx, is the partial integral counterpart to which rule?

Option a: Product Rule: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x). This corresponds to how the derivative of a product expands, and integration by parts is essentially derived from rearranging this product rule. Thus, this option correctly identifies the related fundamental rule.

Option b: Addition Rule: (f(x) + g(x))' = f'(x) + g'(x). This describes the derivative of a sum, which is not directly the rule governing integration by parts. It does not capture the product structure that integration by parts exploits.

Option c: Quotient Rule: (f(x)/g(x))' = [f'(x)g(x) − f(x)g'(x)] / [g(x)]^2. Even though quotient differentiation involves a product of f and 1/g, this option incorrectly states the familiar form and is not the rule that underpins the integration-by-parts identity presented. It confuses a different differentiation rule with a derivative of a ratio.

Option d: Chain Rule: (f(g(x)))' = f'(g(x)) · g'(x). The chain rule handles composition of functions, not the product structure that is central to integration by parts. It does not reflect the integration counterpart shown.

Option e: Power Rule: (f(x)^{g(x)})' = f(x)^{g(x)} [ g'(x) ln(f(x)) + g(x) f'(x)/f(x) ]. This is a generalization for variable exponents and is far afield from the standard product-based integration-by-parts relation. It misapplies differentiation rules to a nonstandard form.

In summary, option a accurately reflects the product rule, which is the foundational rule whose partial integral counterpart leads to the integration-by-parts formula. The other options describe distinct differentiation rules and do not align with the product-structure reasoning behind integration by parts.

Integration by parts [math: ∫f′(x)g(x)dx=f(x)g(x)−∫f(x)g′(x)dx]\int f'(x) g(x) \, dx = f(x) g(x) - \int f(x) g'(x) \, dx is the partial integral counterpart to which of the following rules?

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