I have a rather complex expression $F(x,y)=0$, which implicitly defines $y$, and can find out how $y$ response to a marginal change in $x$ via the implicit function theorem: $$\frac{dy}{dx}=-\frac{F_x}{F_y}.$$ However, I know that $\frac{dy}{dx}$ is negative for low values of $x$, and positive for high values of $x$. Now I want to know the aggregate effect of a discrete change from $x_0$ to $x_1$. Is there an easy way to compute this, or at least figure out whether the discrete change $\frac{\Delta y}{\Delta x}$ is positive or negative?
One way to do it would be to calculate $$\frac{\Delta y}{\Delta x}=\int_{x_0}^{x_1} -\frac{F_x}{F_y} dx, $$ but given how large the expression is, this is probably not tractable.