One formula for "how fast, right here" — and the tangent line it draws.
In Lesson 36.2 you watched a secant line tip over into a tangent as the gap Δx shrank to nothing. That limiting slope earned a name and a symbol: f′(x₀), the derivative of f at x₀. This lesson pins it down. We write the whole shrinking process as a single formula, work it out by hand for f(x) = x² and watch it land on exactly 2x₀, and read its geometry — the derivative is the slope of the tangent line, so point-slope hands us the tangent's equation. Then comes the leap: collect the slope at every point and f′ stops being one number and becomes a whole new function. Finally a caution — every differentiable curve is continuous, yet a sharp corner like y = |x| is continuous and still has no derivative, because the slope coming from the left disagrees with the slope coming from the right.
The average rate of change of f over the stretch from x₀ to x₀ + Δx is the slope of the secant joining those two points:
average rate = f(x₀ + Δx) − f(x₀)Δx
That is the difference quotient. Lesson 36.2's whole story was: don't stop at one Δx — let it shrink. The number the quotient heads toward, as Δx → 0, is the instant rate of change. We give it a name:
f′(x₀) = limΔx → 0 f(x₀ + Δx) − f(x₀)Δx
Build the difference quotient and simplify before letting Δx vanish:
(x₀ + Δx)² − x₀²Δx = x₀² + 2x₀Δx + Δx² − x₀²Δx = 2x₀Δx + Δx²Δx = 2x₀ + Δx
The Δx in the bottom cancelled — no more 0/0. Now let Δx → 0: the leftover Δx melts away and we are left with f′(x₀) = 2x₀. So f′(2) = 4, f′(−1) = −2, f′(0) = 0. The slope of x² at any point is exactly twice the x-coordinate.
Notice the trick: we never plugged in Δx = 0 (that would be 0/0). We simplified first, then took the limit. That two-step move — algebra, then limit — is the heart of every derivative computed from the definition.
Here is the geometry. Pick a point on the curve and zoom in. The more you magnify, the straighter the curve looks around that point — until, very close in, the curve is indistinguishable from a single straight line. That line is the tangent, and its slope is f′(x₀). This is called local linearity: under enough magnification, a smooth curve is its tangent.
Analysis: f′(x₀) is the limit of the difference quotient — the instant rate of change. Geometry: f′(x₀) is the slope of the tangent line to y = f(x) at x₀. They are the same number, seen two ways.
Once you know the slope, the line is easy. A tangent passes through the point of contact (x₀, f(x₀)) with slope f′(x₀). Feed those into point-slope form, y − y₀ = m(x − x₀):
y = f′(x₀)·(x − x₀) + f(x₀)
Point of contact: (3, f(3)) = (3, 9). Slope: f′(3) = 2·3 = 6. Point-slope:
y = 6(x − 3) + 9 = 6x − 18 + 9 = 6x − 9.
Check the contact point: at x = 3, y = 6·3 − 9 = 9. ✓ The line just kisses the parabola at (3, 9).
So far f′ has been one number at one point. But there is nothing special about that point — every x has its own tangent, its own slope. Collect the rule "x ↦ slope of f at x" and you have built a brand-new function, the derivative function f′(x). For f(x) = x², that rule is f′(x) = 2x — a straight line.
Plotting f and f′ on the same axes makes the link visible. Where f is rising, its slopes are positive, so f′ sits above the x-axis. Where f is falling, f′ is below. Where f turns flat (the bottom of the parabola), f′ crosses zero. The height of the green curve at any x is exactly the steepness of the blue curve there.
Differentiation takes a function f and returns another function f′. Feed f′ to the machine again and you get f″ (the second derivative) — the rate at which the slope itself changes. The derivative is a verb you can repeat.
If a curve has a tangent at x₀, it certainly can't have a break there — a jump or a hole would make the secants fly apart instead of settling on one line. So differentiable ⟹ continuous: having a derivative is the stronger condition.
The arrow does not reverse. A curve can be perfectly continuous — drawn without lifting the pen — and still have no derivative at a point. The textbook case is a corner: y = |x| at x = 0. Approach the corner from the left and every secant has slope −1; approach from the right and every secant has slope +1. The two one-sided limits disagree, so there is no single slope, no single tangent, and f′(0) does not exist — even though the function is continuous at 0.
Continuous does not imply differentiable. The corner of |x| is continuous yet has no derivative at 0 — drawing without lifting the pen is not enough; the slope must also settle on one value from both sides.
And the tangent is the limit of secants, not merely "a line that touches at one point." A tangent can cross the curve (the tangent to y = x³ at the origin lies through the curve), while a line that grazes a corner at a single point is not a tangent at all.
| Idea | What it says |
|---|---|
| Definition | f′(x₀) = limΔx→0 [f(x₀+Δx) − f(x₀)] / Δx |
| Worked from scratch | For f = x²: quotient = 2x₀ + Δx → f′(x) = 2x |
| Geometry | f′(x₀) = slope of the tangent line at x₀ (local linearity) |
| Tangent equation | y = f′(x₀)·(x − x₀) + f(x₀) (point-slope) |
| f′ is a function | collect the slope at every x; f rising ⟺ f′ > 0 |
| Smoothness | differentiable ⟹ continuous; a corner (|x|) is the counterexample |
Six questions to lock it in. Tap the answer you think is right.
This lesson gives the formal definition behind the derivative — f′(x₀) as the limit of the difference quotient — and its geometry as the slope of the tangent line, building on HSF-IF.B.6 (rate of change over an interval) toward the instantaneous rate. Computing f′ from the definition and assembling the tangent via point-slope draws on HSF-BF (building and interpreting functions). The idea that f′ is itself a function, and the corner of |x| showing that continuity does not imply differentiability, are precisely the conceptual foundations of AP Calculus AB derivatives. A good follow-up question at home: "If the slope is positive everywhere, can the function ever come back down?"