Question

determine whether the following statement is true or false, and explain why. the derivative of f(x) represents the instantaneous rate of change of y = f(x) with respect to x. choose the correct answer below. a. the statement is false because the derivative is the average rate of change of x with respect to y = f(x). b. the statement is true because the definition of the instantaneous rate of change for a function f at x is the same as the definition of the derivative of a function f at x. c. the statement is false because the derivative is the instantaneous rate of change of x with respect to y = f(x). d. the statement is false because the derivative is the average rate of change of y = f(x) with respect to x.

Explanation:

The derivative of a function $y = f(x)$ represents the instantaneous rate of change of $y$ with respect to $x$, not the average rate of change. The average rate of change is calculated over an interval, while the derivative is the limit as the interval size approaches zero.

Answer:

A. The statement is false because the derivative is the average rate of change of $y = f(x)$ with respect to $x$.