I was playing around and did this:

$\displaystyle f(x)g(x)=f_xg_x$

Then:

$\displaystyle \frac{d}{dx}\left [f_xg_x\right ]=\frac{g_xdf_x}{dx}+\frac{f_xdg_x}{dx}$

so

$\displaystyle f_xg_x=\int \frac{g_xdf_x}{dx}+\frac{f_xdg_x}{dx}dx=\int \frac{g_xdf_x}{dx}dx+\int \frac{f_xdg_x}{dx}dx=\int g_xdf_x+\int f_xdg_x$

so

$\displaystyle f_xg_x=g_x \int df_x+f_x \int dg_x$

and

$\displaystyle f_xg_x=f_xg_x+f_xg_x=2f_xg_x$

Any obvious division by zero someone should point out?

Or do I just not understand the concept of something?