Little question about pattern matching of x_^n_

Question

I'm now practicing pattern matching by implementing a self-defined differentiation operator.

ClearAll[diff]
diff[fx_ + gx_, x_] := diff[fx, x] + diff[gx, x]
diff[c_*fx_, x_] /; FreeQ[c, x] := c*diff[fx, x]
diff[x_^n_., x_] /; FreeQ[n, x] := n*(x^(n - 1))

I test it with

diff[9^a, x]
diff[3, x]

But these two expressions do not further evaluate to the "correct" result. Where is the mistake?

Answer 1

Of the two examples you show, neither of them matches any of the diff patterns you declared:

• In the case of diff[9^a, x], note that your last pattern (diff[x_^n_., x_] /; FreeQ[n, x]) requires that the base of the first argument and the second argument both be x, so diff[x^a, x] or diff[9^a, 9] would match, but not diff[9^a,x].
• In the case of diff[3,x], you have no form of diff that matches on a single value in the first argument; you'd need something like diff[q_,x_] possibly with a requirement like diff[q_?NumericQ, x_]. This case does not match your third diff declaration either because 3 is not x.