The problem is a rather famous (and natural) one. It was first explicitly stated by W. H. Young in 1911. I like quoting this passage so much (even if only to find an excuse to use the word "mooted") that I won't resist that impulse here.
"Recent research [of Lebesgue and Vitali] has provided us with a set of
necessary and sufficient conditions that a function may be the
indefinite integral..., of another function and the way has thus been
opened to important developments. The corresponding, much more
difficult, problem of determining necessary and sufficient conditions
that a function may be a differential coefficient, has barely been
mooted; indeed, though we know a number of necessary conditions no set
even of sufficient conditions has to my knowledge ever been
formulated, except that involved in the obvious statement that a
continuous function is a differential coefficient. The necessary
conditions in question are of considerable importance and interest.
A function which is a differential coefficient has, in fact, various
striking properties. It must be pointwise discontinuous with respect
to every perfect set; it can have no discontinuities of the first
kind; it assumes in every interval all values between its upper and
lower bounds in that interval; its value at every point is one of the
limits on both sides of the values in the neighbourhood; its upper and
lower bounds, when finite, are unaltered, if we omit the values at any
countable set of points; the points at which it is infinite form an
inner limiting set of content zero. From these necessary conditions
we are able to deduce much valuable information as to when a function
is certainly not a differential coefficient . . . . These conditions
do not, however, render us any material assistance, even in answering
the simple question as to whether the product of two differential
coefficients is a differential coefficient, and this not even in the
special case in which one of the differential coefficients is a
continuous function." ...from W H Young, A note on the property of being a differential coefficient. Proc. London Math. Soc. 1911
(2) 9, 360-368.
In the monograph by Andrew M. Bruckner, Differentiation of real functions, Chapter seven there is a discussion of the problem of characterizing derivatives.
Andy has updated his account of this problem in a survey article for the Real Analysis Exchange:
Bruckner, Andrew M. The problem of characterizing derivatives
revisited. Real Anal. Exchange 21 (1995/96), no. 1, 112--133.
Here are a few actual characterizations of derivatives:
C. Neugebauer, Darboux functions of Baire class 1 and derivatives,
Proc. Amer. Math. Soc., 13 (1962), 838–843.
D. Preiss and M. Tartaglia, On Characterizing Derivatives, Proceedings
of the American Mathematical Society, Vol. 123, No. 8 (Aug., 1995),
Chris Freiling, On the problem of characterizing derivatives, Real
Analysis Exchange 23 (1997/98), no. 2, 805-812.
Brian S. Thomson, On Riemann Sums, Real Analysis Exchange 37
My guess is that you won't find any of these satisfying in the way, for example, that "continuous" functions have multiple necessary and sufficient characterizations, most of them natural and compelling.