Let's try

    pow x 0 = 1
    pow x n = x * pow x (n-1)
      ergo
    pow' x 0 = 0
    pow' x n = pow x (n-1) + x * pow' x (n-1)
      ergo
    pow' x 1 = pow x 0 + x * pow' x 0
             = 1 + x * 0
             = 1
    pow' x 2 = pow x 1 + x * pow' x 1
             = x + x
             = 2 * x
    pow' x 3 = pow x 2 + x * pow' x 2
             = x * x + 2 * x * x
             = 3 * x * x

If I understand you correctly: yes. You differentiate both sides and then get an equation for the derivative.

Isn't differentiation itself a recursively defined function?

yes. it generally won't have a closed form however.

Yes. You can also integrate a recursively defined function(s). E.g., recall your undergraduate differential equations course.