Does the left shift operator, as defined here, satisfy an analogue of the product rule, and if so, what? Define the left shift operator as follows:
$$\lambda_x y = \frac{y-(x:=0)y}{x}$$
where $(x:= 0)$ means "replace every copy of $x$ by $0$," or equivalently "evaluation at $0$."
For example,
$$\lambda_x(1+3x-x^2) = \frac{1+3x-x^2-(1+3\cdot 0-0^2)}{x} = 3-x.$$
For comparison, lets compute the derivative, too:
$$\partial_x(1+3x-x^2) = 3-2x.$$
So when $\lambda_x$ is applied to a power series, the constant term is dropped, and the whole series shifts to the left; similar to how differentiation works, except that $\partial_x$ also introduces a modification to the coefficients equal to the exponent of the relevant term. 
Now it's clear that $\lambda_x$ is linear. And, given that $\lambda_x$ is similar to $\partial_x$, we might expect some kind of product rule to hold. But I've been staring at the expression $$\lambda_x(y\cdot z)$$ for awhile, and I'm not seeing anything.

Question. Does the left shift operator, as defined here, satisfy an analogue of the product rule, and if so, what?


$$f(x)g(x)-f(0)g(0)=(f(x)-f(0))g(x)+f(0)(g(x)-g(0))$$
Thus your left shift operator satisfies the Leibniz formula
$$\lambda_x(yz)=(\lambda_xy)z+((x:=0)y)(\lambda_xz)$$
