Integration by parts $$\int f'(x) g(x) \, dx = f(x) g(x) - \int f(x) g'(x) \, dx $$ is the partial integral counterpart to which of the following rules?

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Question restatement: The partial integral counterpart to the formula ∫ f'(x) g(x) dx = f(x) g(x) − ∫ f(x) g'(x) dx is being asked.
Option a: Product Rule: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x). This rule describes how to differentiate a product of two functions and yields the sum of two terms, which directly leads to the integration-by-parts formula when you integrate f'(x)g(x) and rearrange under the integral sign. This aligns with the idea that integration by parts comes from reversing the product rule.
Option b: Addition Rule: (f(x) + g(x))' = f'(x) + g'(x). This rule describes differentiation of a sum and does not involve a product, so it does not underpin integration by parts, which relies on products of functions.
Option c: Quotient Rule: (f(x)/g(x))' = [f'(x)g(x) − f(x)g'(x)] / [g(x)]^2. While structurally related to differentiation of quotients, it does not directly yield the ∫ f'(x) g(x) dx = f(x) g(x) − ∫ f(x) g'(x) dx form as its integration-by-parts counterpart. The quotient rule is a separate rule for derivatives of ratios, not a direct foundation for integration by parts.
Option d: Chain Rule: (f(g(x)))' = f'(g(x)) · g'(x). This rule addresses differentiation of composed functions and is not the source of the integration-by-parts formula, which involves a product of a function and the derivative of another function, not a chain of inner and outer functions.
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 the derivative of an exponential with variable exponent and is not connected to the product rule or the derivation of integration by parts.

Summary: Among the given options, the one that corresponds to the product rule, whose rearrangement under integration leads to the integration by parts formula, is the option describing the Product Rule. The other rules describe different differentiation scenarios (sum, quotient, chain, and variable-exponent powers) and do not serve as the partial integral counterpart to the stated integration by parts formula.

MTH1030 -1035 - S1 2025 MTH1030/35 Week 11 lesson quiz: Differential equations

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

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