Leibnitz's Theorem.—If $n\in\mathbb{N}$,
$$\frac{\mathrm{d}^n}{\mathrm{d}x^n}a(x)b(x) = \sum_{k=0}^{n}\binom{n}{k}a^{(n-k)}(x)b^{(k)}(x)$$
How was this derived?
I know there is a proof by induction but I would like to know Leibnitz steps for the discovery of this equation and some intuition.
If you simply multiply the Taylor series for $y$ and $z$ and collect like powers you quickly get a result that is equivalent to the Leibniz formula. Since Isaac Newton used power series as one of his main tools in his exposition of calculus it would seem likely that Leibniz would know such tricks too.