Linear Algebra
10.108 min read

Rate of Change

The average rate of change of ff between x0x_0 and x0+hx_0 + h is f(x0+h)−f(x0)h\frac{f(x_0+h) - f(x_0)}{h}. This is the slope of the secant line joining (x0,f(x0))(x_0, f(x_0)) and (x0+h,f(x0+h))(x_0+h, f(x_0+h)). It tells us: on average, how much does ff change per unit increase in xx over the interval [x0,x0+h][x_0, x_0+h]?

As h→0h \to 0, the secant line approaches the tangent line at x0x_0. The slope of the tangent line is the instantaneous rate of change — the derivative. This passage from "average rate of change" to "instantaneous rate of change" is the central idea of differential calculus.

Formal View

Definition 10.9 — Average and Instantaneous Rate of Change
The average rate of change of ff over [x0,x0+h][x_0, x_0+h] is
g(h)=f(x0+h)−f(x0)h,h≠0.g(h) = \frac{f(x_0+h) - f(x_0)}{h}, \quad h \neq 0.
The instantaneous rate of change (derivative) at x0x_0 is f′(x0)=lim⁡h→0g(h)f'(x_0) = \lim_{h \to 0} g(h), if this limit exists.

Interactive Visualization

Secant → Tangent Line

Why This Matters

The derivative as a rate of change appears in physics (velocity), economics (marginal cost), and data science (gradient).

  • Physics: velocity v(t)=lim⁡h→0x(t+h)−x(t)hv(t) = \lim_{h\to 0} \frac{x(t+h)-x(t)}{h} is the instantaneous rate of change of position.
  • Economics: marginal cost =ddqCost(q)= \frac{d}{dq}\text{Cost}(q) is the rate of change of cost with respect to quantity.
  • Finance: the delta of an option =∂V∂S= \frac{\partial V}{\partial S} is the rate of change of option price with stock price.

Quiz

Question 1

The average rate of change f(x0+h)−f(x0)h\frac{f(x_0+h) - f(x_0)}{h} is the slope of:

Question 2

As h→0h \to 0, the secant line approaches the tangent line at x0x_0.

Common Mistakes

  • Confusing average rate of change (secant slope) with instantaneous rate of change (tangent slope/derivative).
  • Evaluating at h=0h = 0 directly: g(0)=0/0g(0) = 0/0 is undefined — you must take the limit.