how is integration by parts derived from the product rule for derivatives?

Topic 20 - Explain how Integration by Parts is derived from the Product Rule for Derivatives. First, let's recall what the product rule is. The product rule is when we want to take the derivative of a function that is the product of two functions, as shown below. &lf (x) 9(2)] The way we express this is by taking the derivative of one of these functions times the other function and adding it to the derivative of the other function times the other function, as shown below. + If (x) 9 (2)] = f (x)9(2) + f (2)9 (2) Now to show how integration by parts is related, let's start by taking the antiderivative of both sides of the equation (ignore adding the constant). f(x)g(x) = ff (x) g(x) da + ff(x)g (x) da From here, we want to subtract f f' (x) g (x) da from both sides. f(x) g(z) - ff (x) g(x) dx = ff(x)9 (x) dx Rewritten as: ff(x) g (x) da = f(x)g(x) - ff (x) 9(2) da And, as a reminder, here is an outline of how to apply the Integration by Parts formula in practice: 1. Identify the two factors in the integrand: u and du 2. Find du and v = J du 3. Find fvdu