Derivation of Fick's first law for discrete spaces

Question

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Fick's first law for discrete spaces was explained in the book about biomathematics.

$$ J = -D (C_{i+1} - C_i) $$

In considering the derivation of this equation, I decided to follow the method of deriving Fick's law in continuous space.

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$N$ : Number of particles, $S$ : cross-sectional area.
I attempted to derive the equation based on the number of particles passing through the red filter per unit time and unit area as shown in the figure.

1. $J = - \frac{1}{2S\Delta t} (N_{i+1} - N_i)$
2. $J = - \frac{(\Delta x)^2}{2\Delta t} \frac{1}{\Delta x} (\frac{N_{i+1}}{S \Delta x} - \frac{N_i}{S \Delta x})$
3. $J = -D \frac{C_{i+1} - C_i}{\Delta x}$

$C$ : Volume density.
In the third equation, I want to eliminate $\Delta x$ and derive $J = -D (C_{i+1} - C_i)$. What should I do?