Lesson 4: The Derivative – Instantaneous Rate of Change

1. From Average to Instantaneous Rate

The average rate of change over an interval [a, b] is the slope of the secant line:

\[ \text{Average rate} = \frac{f(b) - f(a)}{b - a} \]

As the interval gets smaller \((h \rightarrow 0)\), this becomes the **instantaneous rate** — the slope of the tangent line at a point. This is the **derivative**.

Definition:

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

If this limit exists, \(f\) is differentiable at \(x\).

2. The Derivative as a Function

\(f'(x)\) gives the instantaneous rate (slope) at every \(x\).

Example: \(f(x) = x^{2}\)

Using the limit definition:

\[ f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = 2x \]

So \(f'(x) = 2x\) — slope doubles as \(x\) increases.

Summary – Lesson 4

• Derivative = instantaneous rate of change = slope of tangent line
• Limit definition: \(f'(x) = \lim_{h\rightarrow 0} [f(x+h) - f(x)] / h\)
• \(f'(x)\) gives slope at every point x

Next: easy rules to compute derivatives without limits every time.