Regression with interaction term that is a function of other regressors Consider the simple interaction equation where moderator M determines the effect of X on Y.
$$
Y=\beta_1X+\beta_2M+\beta_3XM \quad (1)
$$
then the slope on X is
$$
\frac{dY}{dX}=\beta_1+\beta_3M \quad (2)
$$
Fine.
Now suppose that $M=X/Z$. 
You might say that M is just a linear transformation of X, but if X and Z are different for every observation in a data set, M provides some additional information beyond X (I think).
But now our interaction equation is
$$
Y=\beta_1X+\beta_2\frac{X}{Z}+\beta_3\frac{(X^2)}{Z} \quad (3)
$$
and the slope on X is
$$
\frac{dY}{dX}=\beta_1+\beta_2\frac{1}{Z}+\frac{(2\beta_3X)}{Z} \quad (4)
$$
which is very different than in eq (1).
My (maybe ill formed) question is whether it is even acceptable to formulate a regression in this way, and what the implications on interpretation will be. I have a fairly well-formed theory that goes along with the equation, but I just can't seem to make sense of the math.
Any insights are appreciated.
 A: Let me rephrase your question a little bit. The original regression $(1)$ is
$$
Y=\beta_1 X +\beta_2M+\beta_3XM+\varepsilon,
$$
where $\varepsilon$ is the error term. To study the partial effect of $X$ on $Y$, one should look $\mathrm{E}(Y\mid X)$. Assuming $\mathrm{E}(\varepsilon \mid X)=0$, we have
$$
\mathrm{E}(Y \mid X)=\beta_1X+\beta_2\mathrm{E}(M\mid X)+\beta_3X\mathrm{E}(M\mid X),
$$
$$
\frac{\mathrm{d}\mathrm{E}(Y \mid X)}{\mathrm{d}X}=\beta_1+\beta_2\frac{\mathrm{d}\mathrm{E}(M \mid X)}{\mathrm{d}X}+\beta_3\mathrm{E}(M\mid X)+\beta_3 X\frac{\mathrm{d}\mathrm{E}(M \mid X)}{\mathrm{d}X}.\quad(1)
$$
Here (1) reconciles with your (1) and (4) depending on the relationship between $M$ and $X$. If $\mathrm{d}\mathrm{E}(M\mid X)/\mathrm{d}X=0$, we have $(1)$ in your question.
Now let's go back to your original question. It is fine to have a regression like (2) in your post. Under your setting, the partial effect of $X$ on $\mathrm{E}(Y\mid X)$ will be a function of $X$, i.e. the effects would depend on the value of $X$. In particular, we have
$$
\frac{\mathrm{{d}\mathrm{{E}}}\left(Y\mid X\right)}{\mathrm{{d}}X}=\beta_{1}+\beta_{2}\mathrm{E}\left(\frac{1}{Z}\mid X\right)+\beta_{2}X\frac{\mathrm{d}\mathrm{E}\left(1/Z\mid X\right)}{\mathrm{d}X}+2\beta_{3}X\mathrm{E}\left(\frac{1}{Z}\mid X\right)+\beta_{3}X^{2}\frac{\mathrm{d}\mathrm{E}\left(1/Z\mid X\right)}{\mathrm{d}X}.
$$
So to determine the partial effects, you also need to know the conditional mean of $1/Z$ upon $X$.
