Ohio Assessments for Educators (OAE) Mathematics Practice Exam

In the product rule for derivatives, which expression is used?

1. First times the second

2. First times the derivative of the second plus the second times the derivative of the first

3. Derivative of the first times the derivative of the second

4. High times low minus low times high

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

In the product rule for derivatives, the correct expression is derived from the need to find the derivative of the product of two functions. Specifically, if you have two functions, say \( f(x) \) and \( g(x) \), the product rule states that the derivative of their product \( f(x)g(x) \) is given by the sum of the first function times the derivative of the second function plus the second function times the derivative of the first function. Mathematically, this is expressed as: \[ (fg)' = f' g + f g' \] This approach takes into account how both functions interact with each other when differentiating their product. By extending this rule, we ensure that we are accurately capturing the rates of change from both functions involved in the product. Understanding this rule is crucial because it allows for differentiation in more complex scenarios, making it a foundational concept in calculus, particularly when dealing with multiplication of functions. Other methods or rules mentioned, such as taking the product directly or solely focusing on the derivatives without maintaining the product structure, do not fully encompass the nature of differentiation in this context.