The Derivative: Definition and Interpretations

The Derivative: Definition and Interpretation

By the end of this lesson, you will be able to define the derivative as a limit, compute derivatives from the definition, and interpret the derivative as a rate of change and slope of the tangent line.

Two Interpretations of the Derivative

The derivative of a function at a point has two equivalent geometric and physical interpretations:

1. Geometric: The slope of the tangent line to the curve at that point
2. Physical: The instantaneous rate of change of the function at that point

These are the same thing described differently. Both are captured in the definition.

From Average to Instantaneous Rate of Change

Average rate of change over [a, b]:

$$\frac{f(b) - f(a)}{b - a}$$

This is the slope of the secant line connecting the two points (a, f(a)) and (b, f(b)).

As b approaches a, the secant line approaches the tangent line — and the average rate of change approaches the instantaneous rate of change.

The Limit Definition of the Derivative

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Here, h represents the width of the interval approaching 0.

Alternate form (limit as x → a):

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Both forms are equivalent and appear on the AP exam.

Computing Derivatives from the Definition

Example 1: Find f'(x) for f(x) = x²

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

Expand: (x+h)² = x² + 2xh + h²

$$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h}$$

$$= \lim_{h \to 0} (2x + h) = 2x$$

So f'(x) = 2x. This confirms the power rule.

Example 2: Find f'(a) for f(x) = 1/x at x = a

$$f'(a) = \lim_{h \to 0} \frac{\frac{1}{a+h} - \frac{1}{a}}{h}$$

Combine fractions:

$$= \lim_{h \to 0} \frac{\frac{a - (a+h)}{a(a+h)}}{h} = \lim_{h \to 0} \frac{-h}{ha(a+h)} = \lim_{h \to 0} \frac{-1}{a(a+h)} = -\frac{1}{a^2}$$

Differentiability

A function f is if f'(a) exists — that is, if the limit defining f'(a) exists.