The Problem With Instantaneous Speed

Average speed is easy: distance divided by time. Drive 120 miles in 2 hours, average speed is 60 mph. But your speedometer right now reads something specific — say 73 mph. That number describes your speed at one instant, not over an interval. How do you calculate it mathematically?

You cannot divide distance by time for a single instant. At one instant, zero distance passes in zero time: 0/0. That is meaningless as written. So how does the speedometer work?

This is not a new problem. It puzzled Zeno of Elea in 450 BC and remained unsolved for two thousand years until Newton and Leibniz independently figured it out in the 1660s-1670s. Their answer was the derivative.

The Core Idea: Slopes of Secants Approaching a Tangent

Take a curve y = f(x). Draw a chord from the point (x, f(x)) to a nearby point (x+h, f(x+h)). The slope of this chord is:

[f(x+h) − f(x)] / h

This is the average rate of change over the interval from x to x+h. Now shrink h toward zero. The chord rotates, its slope changing as the second point slides toward the first. In the limit — as h actually reaches zero — the chord becomes the tangent line. Its slope is the instantaneous rate of change. That limiting slope is the derivative.

The Formal Definition

f'(x) = limh→0 [f(x+h) − f(x)] / h

If this limit exists, f is differentiable at x and f'(x) is its derivative there. If the limit does not exist — because the function has a corner, a cusp, or a vertical tangent — the derivative does not exist at that point.

Computing From the Definition

📋 Find f'(x) for f(x) = x² from the definition
Write outf'(x) = lim(h→0) [(x+h)² − x²] / h
Expand= lim(h→0) [x² + 2xh + h² − x²] / h = lim(h→0) [2xh + h²] / h
Factor h= lim(h→0) h[2x + h] / h = lim(h→0) (2x + h)
Limit= 2x. So (x²)' = 2x. The slope of x² at any point x is 2x.
📋 Find f'(x) for f(x) = 1/x from the definition
Write outlim(h→0) [1/(x+h) − 1/x] / h
Common denominator= lim(h→0) [x − (x+h)] / [hx(x+h)] = lim(h→0) [−h] / [hx(x+h)]
Cancel h= lim(h→0) −1/[x(x+h)] = −1/x²

Three Ways to Write It

You will see all three notations in textbooks and they all mean the same thing:

f'(x) — Lagrange notation (most common in textbooks) dy/dx — Leibniz notation (emphasises the ratio of changes) Df(x) — operator notation (used in differential equations)

Leibniz notation dy/dx is especially useful because it reminds you what you are doing: measuring how much y changes (dy) per unit change in x (dx). It also makes the chain rule look like fraction cancellation: dy/dx = (dy/du)·(du/dx).

Geometric Meaning

The derivative f'(a) is the slope of the tangent line to the curve at x = a. The equation of that tangent line is:

y − f(a) = f'(a)(x − a)

This is one of the most useful formulas in calculus. Linear approximation, Newton's method, implicit differentiation, and related rates all use tangent lines as their core tool.

When the Derivative Does Not Exist

The derivative fails to exist at three types of points:

📋 Non-differentiable points
Corner|x| at x=0. Left slope is −1, right slope is +1. The limit from both sides disagrees.
Cuspx^(2/3) at x=0. The slope approaches ±∞ from both sides.
Vertical tangentx^(1/3) at x=0. The tangent line is vertical: slope is ∞.

Continuity does not guarantee differentiability. |x| is continuous everywhere but not differentiable at 0. The Weierstrass function is continuous everywhere but differentiable nowhere — a counterintuitive result that shocked 19th-century mathematicians.

Derivatives as Rates of Change

In physics, the derivative of position is velocity: v(t) = ds/dt. The derivative of velocity is acceleration: a(t) = dv/dt = d²s/dt². In economics, the derivative of total cost is marginal cost: MC = dC/dq — the cost of producing one more unit. In biology, the derivative of a population function gives the growth rate. In every case, the derivative measures how fast something changes.

Frequently Asked Questions

Why is the definition written with h→0 rather than h=0?▾
Because at h=0 the expression [f(x+h)−f(x)]/h becomes 0/0 — undefined. The limit process asks what value the expression approaches as h gets arbitrarily close to zero without ever equalling it. This is the fundamental insight of calculus: replacing "at h=0" with "as h→0" makes the problem well-defined.
Do I need to use the definition every time I differentiate?▾
No — and you would go insane if you did. The definition is used once to prove each differentiation rule (Power Rule, Chain Rule, etc.), and those rules are then applied directly. You use the definition to understand what a derivative is and to handle unusual functions not covered by the standard rules.
What does it mean for f'(x) to be negative?▾
It means f is decreasing at that point: as x increases, y decreases. The tangent line slopes downward. If you are measuring position, negative velocity means you are moving backward. If you are measuring revenue, a negative marginal revenue means each additional unit produced reduces total revenue.