Product rule in discrete derivative in finite difference scheme. Suppose we are on real line and I want to discretize the usual derivative operator. Take a smooth function $u$ and step size $h$. Then I could define
$$
\Delta_+u(i) = \frac{u(i+1)-u(i)}{h}
$$
as the discrete form of derivative. Here $i$ is an integer.
Now my question is, when I try to compute $\Delta_+(u(i)\cdot v(i))$, should I have
$$
\Delta_+(u(i)\cdot v(i)) = \frac{u(i+1)v(i+1)-u(i)v(i)}{h} \tag 1
$$
or should I have a sort of product rule and obtain
$$
\Delta_+(u(i)\cdot v(i)) = \Delta_+u(i)\cdot v(i)+u(i)\Delta_+v(i).
$$
Which one is correct? Or which one is better?

The first one is correct by definition. One can try to get a product rule as
$$
\Delta_+(u(i)\cdot v(i))=\frac{u(i+1)v(i+1)-u(i)v(i+1)+u(i)v(i+1)-u(i)v(i)}{h}=\\
=\Delta_+(u(i))\cdot v(\color{red} {i+1})+u(i)\cdot\Delta_+v(i)
$$
but you see the difference in indexing.
