Exploiting Derivative Rules Every diﬀerentiation rule F � (x) = f (x) corresponds to a rule for ﬁnding the anti-derivative F (x) of some function f . a) Find an anti-derivative rule that is the inverse of the sum rule (f + g)� (x) = f � (x) + g � (x). b) Find an anti-derivative rule that is the inverse of the product rule (f ·g)� (x) = f (x)g � (x) + f � (x)g(x). Solution a) Find an anti-derivative rule that is the inverse of the sum rule (u + v)� = u� + v � . If F and G are the anti-derivatives of f and g, respectively, then the anti derivative of: f (x) + g(x) is: F (x) + G(x). We can check this result by diﬀerentiating. b) Find an anti-derivative rule that is the inverse of the product rule (u · v)� = u v � + u� v. If F and G are the anti-derivatives of f and g, respectively, then the anti derivative of: F (x)g(x) + f (x)G(x) is: F (x) · G(x). Later we will study a technique called “integration by parts”, which is closely related to this anti-diﬀerentiation formula. It relies upon the fact that the anti-derivative of F (x)g(x) is equal to F (x)G(x) minus the anti-derivative of G(x)f (x).

Exploiting Derivative Rules

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