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.