The marginal cost function is meaningful only when the produced quantity is normalized The cost function depends on the number of produced items.

Let the value of the cost function C be in hundreds of thousands $ and the quantity is in 1,000,000 items, thus q = 1 indicates 1,000,000 items and q = 0.000001 indicates 1 item.

The marginal cost at a certain number of items (say 1560 items corresponding to q = 0.00156) is in one hand interpreted as the cost of an additional produced item (so it is the cost of the 1561th item), and from another hand it is the derivative of the cost function 

$\frac{\partial \text{C(q)}}{\partial q}$ and then applying q to be 0.00156.

I see a good approximation for $\frac{\partial \text{C(q)}}{\partial q}$ to be $\frac{\partial \text{C(q)}}{1additionalitem}$ since 1 additional item is (0.001561 - 0.001560) which is an infinitly small quantity suitable to be considered as ${\partial q}$.

But what if in some other case, q actually represented just item units and not normalized item numbers e.g.: 

q = 1 indicates 1 item, q = 1560 indicates 1560 produced items, and so on, so in this case $\frac{\partial \text{C(q)}}{\partial q}$ can't be $\frac{\partial \text{C(q)}}{1additionalitem}$ since 1 additional item is (1561 - 1560) which is 1 and not an infinitly small quantity.

would it be possible then to use the derivative of the cost function?

What I'm saying is that considering the marginal cost to be the derivative of the cost function is conditioned by having q a normalized value of a very big number of produced items. Do you agree?
I am not sure I understand your question but if I do, then I do not agree. 

Marginal cost is defined as the derivative of the cost function. If the producible units are too large then marginal cost is less likely be a good approximation of non-marginal cost changes. The importance of this is that the derivative of an estimated cost function will probably not correspond to the real cost increase when non-marginally increasing the output.