The instantaneous rate of change at x for a function f is represented by which expression?

• A. f''(x)
• B. f(x)/x
• C. ∫ f
• D. f'(x)

Answer: (D)

The instantaneous rate of change of f with respect to x is captured by the derivative, which tells how f changes at a single point on the graph. It equals the slope of the tangent line there and is defined as the limit of the average rate of change as the interval around x shrinks to zero: lim_{h→0} [f(x+h) − f(x)]/h. This limiting process is exactly what f'(x) represents, so the expression for the instantaneous rate of change is f'(x).

The other expressions describe different ideas: f''(x) is the rate of change of the rate of change (the second derivative), not the instantaneous change itself; f(x)/x is just a ratio and doesn’t inherently measure how f changes at x; ∫ f denotes accumulation over an interval, not an instantaneous rate at a point.