Is resistance the derivative dV/dI or the ratio V/I?
Ohm's Law is the statement that \$V = I\cdot R\$. Two-terminal devices for which this relation holds are devices that we can model as a resistor.
On the other hand, the distinction between R = V/I and R = dV/dI is one of small-signal versus large-signal model, and is very important to the analysis of analog, linear circuitry such as amplifiers driven with small signals or in feedback.
Consider the following circuit:



simulate this circuit &ndash; Schematic created using CircuitLab
This is an inverting, common-source amplifier. I've neglected biasing details, but suppose that it is biased at a suitable operating point where it acts like an amplifier. We can model the drain current of the MOSFET as
$$I_d = k(V_{in} - V_{th})^2(1 + \lambda\cdot V_{ds})$$
where k and lambda are a consequence of the transistor's construction. For a small signal excitation around our operating point we can take a first-order Taylor expansion and model the drain current instead as
$$ I_d = I_0 + g_m\cdot(V_{in} - V_{in,0}) + g_o\cdot(V_{ds} - V_{ds,0}) $$
corresponding to the following small-signal model:



simulate this circuit
This model now corresponds to the small dI/dV derivatives that you ask about, and is valid for small excitations about the operating point. As you can see, the model draws resistors and linear dependent sources, even though they are non-physical--the MOSFET does not contain a parallel resistor that can be modeled by Ohm's law in the large-signal regime. Because it is linear, we use superposition to combine it with the constant voltages and currents seen at the operating point.
The take-away here is that we have all of our mathematical tools for linear circuits available when we deal with these small-signal models. Even though our MOSFET was hopelessly non-linear, we linearized it and can now do analyses such as finding Thévenin equivalents. We now deal with &quot;resistors&quot; that model the small-signal resistances dV/dI instead.
Looking at the transfer characteristic of our circuit and of the small-signal model, you can see that the name makes sense: it's a good model for small excitations about the operating point, and a poor model for larger excitations. The black trace is the real response, while the green dashed line is that predicted by the small-signal model.