Start with the simple identity: $$(f(x) - a)(g(x) - b) + a(g(x) - b) + b(f(x) - a) = f(x)g(x) - ab.$$

If $a$ and $b$ are the respective values of $f$ and $g$ at some point, then, after dividing both sides by $\Delta x$ and letting $\Delta x\to 0$ the first term vanishes and we get a proof of the product rule.

But if $a$ and $b$ are the respective *average* values of $f$ and $g$, then after averaging the whole expression, the first term is the only one on the left that does *not* vanish and we get the familiar identity $\mathrm{cov}(f,g) = E(fg) - E(f)E(g)$, where $\mathrm{cov}(f,g)$ is by definition the average of the first term.

I don't know that I've ever seen it stated that the product rule and this identity on covariances are the local and global versions of something.

Is this just one instance of some broader pattern?