Here are a few characterizations of derivatives:
On Characterizing Derivatives
D. Preiss and M. Tartaglia On Characterizing Derivatives Proceedings of the American Mathematical Society, Vol. 123, No. 8 (Aug., 1995), pp. 2417-2420.
Chris Freiling, Chris On the problem of characterizing derivatives., Real Anal. Analysis Exchange 23 (1997/98), no. 2, 805–812805-812.
The problem was first posed by W. H. Young. We include in our article about the Youngs a full quote stating his problem;
Bruckner, Andrew M. and Thomson, Brian S. Real variable contributions of G. C. Young and W. H. Young. Expo. Math. 19 (2001), no. 4, 337–358. [You can download the PDF file here.]