How can I use UpSet with a derivative value?

Question

I'm trying to UpSet a certain value for the derivative of a function $Y=Y(l,m,x)$. This can be done by:

D[Y[l_, m_, x_], x_] ^= (l + 1) Y[l, m, x];
D[Y[a, b, c], c]
(*(1 + a) Y[a, b, c]*)

But the problem is, this doesn't always work since for other types of expressions, because Mathematica uses a different form:

D[c *Y[a, b, c], c]
(*Y[a,b,c]+c (Y^(0,0,1))[a,b,c]*)

I've tried solving this with another UpSet, but I get an error.

(Y^(0,0,1))[a_,b_,c_] ^= (l+1) Y[l,m,x]

.

How can I solve this?

Answer 1

Try

Derivative[0, 0, 1][Y][l_, m_, x_] := (l + 1) Y[l, m, x]
D[c*Y[a, b, c], c]
(*Y[a, b, c] + (1 + a) c Y[a, b, c]*)