The derivative is a slope

Pre-calculus ended with the limit. Calculus begins by pointing that idea at one question: how fast is a function changing at a single instant? The answer is the derivative.

Average rate vs. instant rate

The slope between two points on a curve — rise over run — is an average rate of change over that stretch. But the slope at one point? A single point has no run to divide by. The trick is to sneak up on it with a limit.

Fix a point P on the curve and pick a second point Q. The line through them is a secant, and its slope is the average rate from P to Q. Now slide Q toward P: the secant pivots, and in the limit it becomes the tangent line at P. The slope of that tangent is the derivative.

The definition

Written as a limit, the derivative of f at x is:

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

The h is the run between the two points; f(x + h) − f(x) is the rise. Taking h → 0 slides the second point into the first — exactly the picture above.

A worked limit

For f(x) = x², expand the difference quotient:

• [ (x + h)² − x² ] / h
• = [ x² + 2xh + h² − x² ] / h
• = [ 2xh + h² ] / h = 2x + h

Now let h → 0: the leftover h vanishes and f′(x) = 2x. At x = 3 the slope is 6; at x = 0 it's 0 (the bottom of the parabola is flat). No guessing — the limit delivers the exact slope.

Why this matters

"Instantaneous rate of change" is everywhere the world moves: velocity is the derivative of position, current is the derivative of charge, marginal cost is the derivative of cost. Every one of them is this same secant-into-tangent limit.