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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.

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What about solve $F(x_0,y_0)=0$ and $F(x_1,y_1)=0$ for $y_0$ and $y_1$ and after that calculate $\frac{\Delta y}{\Delta x}= \frac{y_1 -y_0}{x_1 -x_0}$? – RicardoCruz Jul 12 '13 at 1:20
I can't solve those terms explicitly unfortunately. – Nameless Jul 12 '13 at 9:26

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