Every calculus student learns, at some point, that the derivative is "the slope of the tangent line." It's a tidy phrase, and it's not wrong — but it's also not a definition. A slope, in the sense you learned in algebra, is rise over run between two points on a line. A curve doesn't have two points that pin down a tangent the way two points pin down a secant. So what, precisely, is the number we call "the derivative," and why does the precise version of the definition occasionally produce a surprising answer: no derivative at all, even at a point where the function is perfectly continuous? Spivak's Calculus spends the opening pages of Chapter 9 building exactly this definition from scratch, and it's worth walking through slowly, because the payoff is a tool that works even where intuition doesn't.
The Problem With "Rise Over Run"
Take two distinct points on the graph of a function f: the point (a, f(a)) and a nearby point (a + h, f(a + h)), where h is just some nonzero number. These two points determine a perfectly ordinary straight line — a secant line — and its slope is the familiar rise-over-run quotient
[f(a + h) − f(a)] / h.
This quotient is well-defined for any h ≠ 0, no matter how strange the curve is at the point in between. The trouble is that this is the slope of a line through two points on the curve — not a line that's tangent to it at a single point. As Spivak observes, a tangent line can't simply be defined as "a line that touches the graph only once": that criterion is both too strict (it would disqualify the tangent line to a wavy curve that happens to cross back through it elsewhere) and too generous (a parabola would then have two "tangent lines" through any point not on it). The honest way forward is to let h shrink toward zero and ask what the secant slopes are doing as they do.
The Actual Definition
This is the move that makes calculus calculus. Instead of picking a single secant line, we look at the whole family of them, indexed by h, and take a limit:
limh→0 [f(a + h) − f(a)] / h
If this limit exists, f is called differentiable at a, the limit itself is denoted f′(a), and the tangent line to the graph at (a, f(a)) is defined to be the line through that point with slope f′(a). Notice what's happened: "tangent line" isn't a primitive geometric idea being described — it's a concept built entirely out of the limit, and it exists only when that limit exists. This is also the definition that makes the physical picture precise: if s(t) gives a particle's position at time t, then [s(a + h) − s(a)] / h is its average velocity over the interval from a to a + h, and s′(a) — the limit of those average velocities — is what we actually mean by "instantaneous velocity," a quantity that, strictly speaking, no physical measurement ever produces directly.
A Worked Example: Differentiating x²
The definition is only useful if you can compute with it, so let's do it for f(x) = x². Plugging straight into the limit:
f′(a) = limh→0 [(a + h)² − a²] / h = limh→0 [a² + 2ah + h² − a²] / h = limh→0 (2a + h) = 2a.
Every step is algebra except the last, where the limit simply evaluates the expression 2a + h at h = 0 (there's no division-by-zero problem left, because the offending h in the denominator canceled out along the way — that cancellation is exactly why the limit exists). The same style of computation shows f(x) = x³ has derivative f′(a) = 3a², and that any linear function f(x) = cx + d has the constant derivative f′(a) = c everywhere — which makes sense, since a line's tangent line at any point is just itself.
Where the Limit Refuses to Exist: f(x) = |x|
Here is the case that shows the definition has real teeth. Consider f(x) = |x| at a = 0. The difference quotient is
[f(0 + h) − f(0)] / h = |h| / h.
For h > 0, |h| = h, so this quotient equals 1. For h < 0, |h| = −h, so the quotient equals −1. The right-hand limit is 1; the left-hand limit is −1. Since these disagree, limh→0 |h|/h simply does not exist, and f is not differentiable at 0 — even though f(x) = |x| is continuous everywhere, including at 0, with no gaps or jumps in its graph at all. Geometrically, the graph has a sharp corner there, and no single line is tangent to both sides of the "V" simultaneously. (Spivak's book pushes this further: f(x) = √|x| is even worse-behaved at 0, since its difference quotient grows without bound rather than merely disagreeing from each side, while a related function like x²sin(1/x) — extended by f(0) = 0 — turns out to be differentiable at 0 after all, with f′(0) = 0, despite oscillating infinitely often near the origin.)
Differentiable Is Stronger Than Continuous
The |x| example is the concrete face of a general theorem: differentiability at a point implies continuity there, but the converse is false. The proof is short — if f is differentiable at a, then
limh→0 [f(a+h) − f(a)] = limh→0 { [f(a+h) − f(a)]/h } · h = f′(a) · 0 = 0,
and that's precisely the statement that f is continuous at a. So every differentiable function is continuous — but as |x| shows, plenty of continuous functions are not differentiable, and the book notes it's possible to build a function that's continuous absolutely everywhere and differentiable absolutely nowhere. Differentiability is a genuinely stronger, more delicate condition than mere continuity, and the limit definition is precisely what lets us tell the two apart instead of just gesturing at a picture of a "smooth" curve.
This is the kind of foundation that makes everything later in calculus — the product rule, the chain rule, optimization, related rates — actually trustworthy rather than merely plausible. If you want to see this definition built up rigorously, one careful step at a time, alongside dozens of worked examples and problems that test whether you really understand where the definition can fail, it's worth spending real time with Spivak's Calculus, 4th Edition.
Further reading: How to self-study Spivak's Calculus · Table of contents · Index · Calculus, 4th Edition · Answer Book
Where else do you think "obvious" geometric intuition — like a sharp corner not having a tangent line — turns out to need this kind of careful proof?
