Given a non-commutative basis ${x_0,x_1,x_2}$ I'd like to define a differential operator that acts as so
$$ \Delta_i (\sum_{n=0}^\infty c_n x_i^n) = \sum_{n=1}^\infty c_n x_i^{n-1}, \quad \Delta_i x_j = 0 \quad\text{if} \quad i\neq j $$
So if this operator was a "derivative" with respect to $x_0$, when it operators on a series in $x_0$ it will just reduce the power of $x_0$ by one. Furthermore I would like to define it so that it specifically acts from the left, such that if I had a string
$$\Delta_0 (x_0 x_0 x_0 x_1 + x_1 x_0 x_2 x_0 + x_2 x_2 x_1) = x_0 x_0 x_1 $$
so if a $\Delta_0$ meets anything that isn't $x_0$ on the left, it will annihilate it.
To generate the non-commutativity I understand that I can just write out the functions I wish to operate on with ** between basis elements. However I am struggling to figure out a way of writing down an operator that matches the properties that I want, described above.
Any help would be greatly appreciated!