Ohio Assessments for Educators (OAE) Mathematics Practice Exam

How is the derivative of a quotient f(x)/g(x) calculated?

1. Low d high plus high d low

2. Low d high minus high d low all above the square of what's below

3. Derivative of the numerator divided by the derivative of the denominator

4. First times the low minus second times the high

The correct answer is: Low d high minus high d low all above the square of what's below

To calculate the derivative of a quotient \( \frac{f(x)}{g(x)} \), the correct approach involves applying the Quotient Rule. The Quotient Rule states that if you have two functions \( f(x) \) and \( g(x) \), the derivative of their quotient is given by the formula: \[ \left(\frac{f(x)}{g(x)}\right)' = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2} \] This means that to find the derivative, you take the derivative of the numerator (low) times the denominator (high) and subtract it from the numerator (high) times the derivative of the denominator (low). The result is then divided by the square of the denominator. Thus, the correct answer captures this process by stating it clearly: you multiply the derivative of the numerator by the denominator, subtract the product of the numerator and the derivative of the denominator, and place this entire expression over the square of the original denominator. Understanding this rule is crucial for solving problems that involve functions divided by one another, as it ensures the right application of derivatives in calculus.