If you know how $u$ changes and how $v$ changes, how does the product $uv$ change? The natural guess is to multiply the changes: $d(uv)=du\cdot dv$. It looks so symmetrical, so elegant… Leibniz made the same guess. Leibniz tests and fails In the same manuscript of 11 November 1675 in which he began using the letter $d$, Leibniz asked himself whether $d(xy)$ would equal $dx\,dy$, and whether $d\!\left(\frac xy\right)$ would equal $\dfrac{dx}{dy}$. Instead of trusting elegance, he tested it with an example and found it did not. A few days later he found the right rule.
A car leaves one city and reaches another, 90 km away, in one hour. Its average speed was 90 km/h. But at every instant the speedometer showed a different number. What exactly does it measure? Speed is distance divided by time, but in an instant no distance is covered and no time passes. Dividing zero by zero gives no answer. That was the question Newton wanted to answer in 1665. His answer was the method of fluxions.
Calculus teachers tend to say two things at the same time: “$\frac{dy}{dx}$ is not a fraction” and “you can cancel the $du$ as in a fraction”. Both are right, and that is precisely the strength and the danger of Leibniz’s notation. It looks like a fraction and, in most calculations, it behaves like one. Knowing where the resemblance holds and where it breaks is one of the subtlest skills in reading mathematics.
A car passes kilometre 120 at 2:00 p.m. and kilometre 210 at 3:00 p.m. Its average speed was $$\frac{\Delta s}{\Delta t}=\frac{210-120}{15-14}=90\ \text{km/h}.$$The Greek letter $\Delta$, capital delta, is the Greek $D$, for difference. $\Delta s$ is not “delta times $s$”: it is a single sign meaning “how much $s$ has changed”. It is a calculation with real numbers, one you can do with a ruler and a clock. But the speedometer does not show the average speed over an hour. It shows the speed now. And “now” has no duration: $\Delta t=0$, and dividing by zero makes no sense. Calculus resolves this impasse with a passage to the limit, and its notation follows the passage without changing shape.
If $y=x^3$, the derivative is $3x^2$. Everyone writes that the same way. What changes is the name given to that derivative. In a physics book you will read $\dot y$; in a calculus book, $\frac{dy}{dx}$ or $f'(x)$; in a book on differential equations, $Dy$. They are four signs with the same object, and none of them has beaten the others. The four ways of writing Notation Author When How it reads $\dot y$ Isaac Newton 1660s, published in 1704 the fluxion of $y$, the speed at which $y$ flows in time $\dfrac{dy}{dx}$ Gottfried Leibniz 1675, published in 1684 the ratio of two infinitely small differences $f'(x)$ Joseph-Louis Lagrange 1797 the derived function of $f$, evaluated at $x$ $Df$ Louis Arbogast 1800 the operator $D$ applied to the function $f$ Chapter 1 has already told the quarrel between Newton’s dots and Leibniz’s $d$s, in the text on Leibniz and the calculus. Here another question matters: what each way of writing shows and what it leaves out.