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.