Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which process applies when differentiating using the product rule?

1. First function times derivative of second function

2. Derivative of the first times the second function

3. Take the square and reduce

4. First times derivative of second plus second times derivative of first

The correct answer is: First times derivative of second plus second times derivative of first

The product rule is a technique used in calculus to differentiate functions that are products of two or more functions. When applying the product rule, you take the derivative of each of the functions involved while also multiplying them appropriately. The correct application of the product rule involves differentiating the first function and multiplying by the second function, and then adding that to the product of the first function and the derivative of the second function. This can be summarized in the formula: if you have two functions, u(x) and v(x), the derivative of their product, u(x)v(x), is given by: \[ \frac{d}{dx}[u(x)v(x)] = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx} \] This reflects the idea that both functions contribute to the rate of change of their product. The two parts that are added together highlight the need to account for how each function affects the other as they vary. Thus, the expression correctly represents the steps you would follow when using the product rule to differentiate.