Limit Definition

The Limit Definition of the Derivative

The derivative captures the idea of instantaneous rate of change by examining what happens when we zoom in infinitely close to a single point.

From Secants to Tangents

Imagine drawing a line (called a secant line) through two points on a curve: $(x, f(x))$ and $(x+h, f(x+h))$. The slope of this secant line is the average rate of change: $\frac{f(x+h)-f(x)}{h}$

As $h$ shrinks toward zero, the second point slides closer to the first, and the secant line rotates into the tangent line—the line that just barely touches the curve at that one point. The slope of this limiting tangent line is the derivative.

The Formal Definition

There are two equivalent ways to express the derivative:

$h$-based form: $f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$

Point-based form: $f'(a) = \lim_{x\to a} \frac{f(x)-f(a)}{x-a}$

Both capture the same limiting slope. The Leibniz notation $\dfrac{dy}{dx}$ emphasizes the ratio of infinitesimal changes.

Example Calculation

For $f(x) = x^2 + 3x$: $f(x+h) - f(x) = 2xh + h^2 + 3h = h(2x+h+3)$ $f'(x) = \lim_{h\to 0}(2x+h+3) = 2x+3$

Differentiability

A function is differentiable at $a$ when this limit exists as a finite number. Differentiability implies continuity, but not vice versa. A continuous function may fail to be differentiable at corners (like $|x|$ at $x=0$), cusps, or vertical tangents.