Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What does the chain rule f(g(x))' involve?

1. f'(g(x)) + g'(x)

2. f'(g(x)) × g'(x)

3. g'(x) - f(g(x))

4. g(f(x)) + f'(x)

The correct answer is: f'(g(x)) × g'(x)

The chain rule is a fundamental technique in calculus for differentiating composite functions. When you have a function that is composed of two functions, such as f(g(x)), the chain rule provides a method to find the derivative of this composite function. The correct choice indicates that the derivative of f(g(x)) can be computed by first taking the derivative of the outer function, f, evaluated at the inner function g(x), which is expressed as f'(g(x)). Then, this result is multiplied by the derivative of the inner function, g(x), represented as g'(x). This relationship is succinctly captured in the rule: (f(g(x)))' = f'(g(x)) × g'(x). Therefore, the correct response reflects the application of the chain rule, emphasizing the multiplication of the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function itself, resulting in a coherent method for differentiating composite functions in calculus.