You just gotta check the action of the basis elements under the map.

For instance the first basis element 1 goes to 0 the derivative of 1 is

So that is your matrix.

First column all zeros

second column has a 1 in the first spot then all 0s

Third column has a 2 in the second spot 0s elsewhere

4th has a 3 in the 3rd spot and 0s elsewhere

that is 1,2,3 on the first superdiagonal.

Check its action on an arbitrary degree 3 polynomial

Looks good to me.