You may go with polynomials over $\mathbf F_2$, the smallest finite field and one of the representations of Boolean functions. Note, that in $\mathbf F_2$ for any $x$ (i.e. $1$ or $0$) the following relations hold:

$$x^2 = x, \quad x + x = 0.$$

The derivative is defined as:

$$\left(D \, f \right)(x) = f(x+1) - f(x)$$

which is also a finite difference. All the usual derivative properties, including the chain rule, are fulfilled. Indeed one can easily show it by studying every possible $f$ --- all the two of them: identity $x \mapsto x$ and negation $x \mapsto x+1$.

Well, considering one-variable functions $\mathbf F_2 \to \mathbf F_2$ is not very interesting and it is better to proceed with multivariable calculus on $\mathbf F_2^N = \underbrace{\mathbf F_2 \times \ldots \times \mathbf F_2}_N $. The directional derivative is defined as

$$\left(D_\boldsymbol u \, g \right)(\boldsymbol x) = g(\boldsymbol x+ \boldsymbol u) + g(\boldsymbol x)$$

where

$$g : \mathbf F_2^N \to \mathbf F_2, \quad \boldsymbol x, \boldsymbol u : \mathbf F_2^N$$

and $+$ is defined as component-wise addition:

$$\boldsymbol x + \boldsymbol u = \left(x_1 + u_1, \ldots , x_N + u_N \right).$$

The chain rule for the above g and $f : \mathbf F_2 \to \mathbf F_2$ is fulfilled:

$$\left(D_\boldsymbol u \; f \circ g \right)(\boldsymbol x) = (D_\boldsymbol u \; f) \, (g(\boldsymbol x)) \cdot (D_\boldsymbol u g)(\boldsymbol x),$$

which again is possible to prove by considering every possible $f$. At least this is how I was able to prove it, and I am not in anyway a mathematician. Unfortunately, a quick search over Google books reveals little (nothing?) on the chain rule for boolean functions. As for me I used "Boolean Functions in Coding Theory and Cryptography" by *O. A. Loginov, A. A. Salnikov, and V. V. Yashchenko* for the introduction to boolean functions, but again it does not go into the chain rule.

I would appreciate if anyone would point me some books or articles on the topic of derivatives of boolean functions, especially when one deals with functions of the type $\mathbf F_2^N \to \mathbf F_2^M$.