Hello. Bonjour. Hola. Three words, three countries, one meaning. Mathematics has done exactly the same thing with the derivative function: the single idea "the rate at which the output changes as the input changes" is written f'(x) in one book, \tfrac{dy}{dx} in the next, and \dot{y} in a third — and they all mean the same thing.

Why three? Because the derivative was invented more than once, by people who cared about different things, and each inventor's notation turned out to be best in its own country. Lagrange's tidy prime f'(x) rules in algebra-land, where you manipulate functions and plug in points. Leibniz's \tfrac{dy}{dx} rules in the land of rates and units — science, economics, anywhere a quantity changes per something. And Newton's dot \dot{y} rules in physics, where the variable is almost always time and you differentiate so often you want the smallest possible mark.

This page makes you fluent in all three dialects. That matters more than it sounds: a physics textbook, a maths exam and an economics paper will each write the same derivative differently, and the reader who only knows one notation keeps mistaking old friends for strangers. By the end you'll translate freely — one idea, three languages.

Lagrange's prime: the algebraist's dialect

The most compact notation is the prime, introduced by Joseph-Louis Lagrange. If y = f(x), its derivative is written:

Read f'(x) as "f prime of x". Its great strength is that it treats the derivative as a function with a name. Once f' is a named function, everything you already do with functions works instantly: evaluate it at a point (f'(3) — no new machinery needed), compose it, graph it, differentiate it again. Higher derivatives just stack primes — f''(x), f'''(x) — and when the primes get silly, a bracketed count takes over: f^{(4)}(x) for the fourth derivative.

The shorthand y' is even terser, but notice what it doesn't say: which variable we differentiated with respect to. With one variable on stage that's harmless; with several it's a trap we'll flag below.

Leibniz's \frac{dy}{dx}: the scientist's dialect

Gottfried Leibniz wrote the derivative as a ratio, deliberately echoing the difference quotient \tfrac{\Delta y}{\Delta x} — "change in y over change in x" — with the capital deltas softened to lowercase d's to say the steps have shrunk to nothing:

Read \tfrac{dy}{dx} as "dee y dee x". Two things make this dialect beloved. First, it names both variables: the top says what is changing, the bottom says with respect to what. Ambiguity is impossible — \tfrac{dV}{dt} and \tfrac{dV}{dr} are visibly different questions about the same volume. Second, the \tfrac{d}{dx} form is an operator — a machine that says "take the derivative with respect to x of whatever follows", as in \tfrac{d}{dx}\,(x^2) = 2x. You can point that machine at any expression without first giving it a name.

And there is a third gift, so useful that scientists essentially refuse to give the notation up: units ride along for free. We'll see it in a moment.

Newton's dot: the physicist's dialect

Isaac Newton got there first (he called derivatives fluxions), and his mark is the smallest of all — a single dot over the letter:

Read \dot{y} as "y dot". The dot carries a hidden convention that you must know to read it: a dot always means the derivative with respect to time. Not with respect to x, not with respect to anything else — time, full stop. So if s(t) is a particle's position, then \dot{s} = \tfrac{ds}{dt} is its velocity, and the double dot \ddot{s} = \tfrac{d^2s}{dt^2} — the derivative of the derivative — is its acceleration.

Why keep such a restricted dialect alive? Because in mechanics everything is differentiated with respect to time, constantly. When an equation contains five time-derivatives, writing \dot{x}, \ddot{x} instead of \tfrac{dx}{dt}, \tfrac{d^2x}{dt^2} keeps it readable. Newton's second law for a spring, m\ddot{x} = -kx, says in six symbols what Leibniz's dialect needs a small fraction tower to say.

Worked example: one sentence, three languages

Take one concrete statement and say it in every dialect. A ball is dropped; after t seconds it has fallen h(t) = 5t^2 metres. The statement we want to write is: "the ball's speed after t seconds is 10t metres per second."

Lagrange names the derivative as a function of t:

Leibniz shows what changes (h, on top) and with respect to what (t, on the bottom):

Newton notices the bottom variable is time — his home turf — and shrinks the whole apparatus to one dot:

Three spellings, one fact. And each earns its keep: h'(t) is the handiest if you next want to study the speed as a function; \tfrac{dh}{dt} is the clearest if a reader might wonder "per what?"; \dot{h} is the fastest to write in a page of mechanics.

Why scientists love Leibniz: the units come free

Here is the third gift promised above. Suppose V is the volume of water in a bath, measured in litres, and t is time in seconds. What are the units of the derivative? In Leibniz notation you barely have to think:

The notation is shaped like "a little bit of V divided by a little bit of t", so the units of the top and bottom divide too. If the tap fills the bath at \tfrac{dV}{dt} = 0.3 litres per second, everyone — plumber, physicist, examiner — knows exactly what that number means. The same trick works everywhere: \tfrac{ds}{dt} in metres per second, \tfrac{dC}{dq} in pounds per unit produced, \tfrac{dP}{dh} in pascals per metre of depth.

Compare the alternatives: V' and \dot{V} are the same quantity, but neither displays its units — you must remember them. This is why lab reports, chemistry, biology and economics overwhelmingly speak Leibniz: the notation doubles as a dimensional sanity check. If your claimed formula for \tfrac{dV}{dt} works out in litres instead of litres per second, you've made an error and the notation itself just caught it.

Evaluating at a point — in each dialect

A derivative is a function, and functions get evaluated. Each dialect has its own way of saying "…and now give me the value at one specific input". Lagrange's is effortless — primes make the derivative a named function, so you evaluate like any other function. Leibniz's \tfrac{dy}{dx} is a symbol, not a name, so it can't take an argument in brackets; instead it hangs a vertical bar off the expression, subscripted with the point:

Read the bar as "evaluated at". For our falling ball at t = 2 seconds, all three dialects report the same 20 metres per second:

One warning about a spelling you might be tempted to invent: the derivative at x = 3 is not written \tfrac{dy}{d3}. The x in dx names the variable you differentiate with respect to; the bar names the value you substitute afterwards. First differentiate, then evaluate — the notation keeps those two acts separate on purpose.

Same idea, side by side

Let's line all three dialects up on one function. Take y = f(t) = t^2 with t playing the role of time (so Newton is allowed to join in). Step through the table: the same derivative, 2t, wearing each country's clothes — and the same value, 6, at the point t = 3.

Notice what stays fixed as the spelling changes: the function, the rule 2t, and the value at the point. Notation is packaging. Exams and textbooks will hand you any of these packages without warning, and expect you to unwrap the same idea from each.

The same slope, however you write it

One picture to pin it all down. Below is y = x^2 with its tangent line at a movable point. Slide x and watch the readout: f'(x), y' and \tfrac{dy}{dx} aren't three numbers that happen to agree — they are three names for the one number the tangent line defines, the slope at that point.

(Newton's \dot{y} sits this one out: here the input is x, a plain number, not time — and the dot dialect is only spoken where time is the variable. That's the point of knowing the conventions: you also know when a notation doesn't apply.)

Watch it explained

Sal Khan tells the story of Newton's and Leibniz's notations — and why, three centuries on, we still happily use both (with Usain Bolt as the moving object).