Is there a way to combine functions so that you combine their derivatives?

Question

Suppose $y,z$ are functions.

What manipulation: "$?$" to the functions would yield the following? (if any)

$$y?z=y\cdot z\\~\\ \frac {d(y?z)}{dx}=\frac{dy}{dx}\cdot\frac{dz}{dx}\\~\\ \frac {d^2(y?z)}{dx^2}=\frac {d^2y}{dx^2}\cdot \frac {d^2z}{dx^2}$$

I already know the chain rule is futile here because after the first derivative you have the product in the second.

Answer 1

It is unclear what you are asking. Why are you defining so many letters?

In any case, Convolutions have a property you may be interested in: https://en.wikipedia.org/wiki/Convolution#Integration, but you may need some technical conditions on the function space for these to work. In particular, as the linked page mentions, you'll need Fubini's theorem or one of it's generalizations to hold.

Answer 2

With the so-called logarithmic derivative, you can write

$$(\log(yz))'=\frac{(yz)'}{yz}=\frac{y'}{y}+\frac{z'}{z}.$$

This does not generalize to the second order.

Answer 3

Utterly impossible: $$y?1=y\cdot 1 = y,$$ $$y' = \frac{d(y?1)}{dx}=\frac{dy}{dx}\cdot\frac{d1}{dx} = 0$$