Proof

Suppose that h is a multiple factor of f. Then there is a polynomial g with f = gh².

The product rule for differentation is valid for D. (Prove it yourself!) Applied to f it yields

Df = (Dg)h² + 2ghDh = ((Dg)h + 2gDh)h.

Consquently, h is a divisor of Df. But then h also divides gcd(f,Df).