Leibniz found the product rule by throwing away a tiny rectangle
How fast does a product of two changing quantities change? Gottfried Leibniz pictured the product as a rectangle's area. Nudge both sides a little and the area grows by two thin strips plus one minuscule corner piece. He declared that corner negligible, and the product rule of calculus fell out.
The rule says that the rate of change of u times v equals the rate of change of u multiplied by v, plus u multiplied by the rate of change of v. It is sometimes called the Leibniz rule, after its usual discoverer, though J. M. Child, a translator of Leibniz's papers, argued the credit belongs to Isaac Barrow. Leibniz reasoned with infinitesimals, quantities smaller than any ordinary number: the new product minus the old one equals u times the small change in v, plus v times the small change in u, plus the product of the two small changes, which he dropped as vanishingly small. Newton reached the same result by thinking about flowing quantities. Neither argument would pass modern standards of rigour.
A modern proof avoids infinitesimals with a sly move. In the limit definition of the derivative, you add and subtract the same cross term in the numerator, which changes nothing but lets the expression split into two familiar pieces. Letting the step shrink to zero, and using the fact that differentiable functions are continuous, gives the rule. Another route uses linear approximations, where every leftover error term shrinks faster than the step itself.
In practice it is a daily tool. Differentiating x squared times the sine of x, for instance, gives 2x times sine of x plus x squared times cosine of x. Treat one factor as a constant and the rule reduces to the constant multiple rule, since a constant's derivative is zero; combined with the sum rule, this shows differentiation is linear.
Much else grows from it. Integration by parts is the product rule run backwards, and a weak form of the quotient rule follows too. It extends naturally to products of three or more functions and to higher derivatives of a product.
Source: Product rule
