For example is we have a function m(x) that describes the distribution of mass along a line, so m(x) gives you the amount of mass at a certain length of the line. If we take the derivative of m with respect to x we get dm/dx or density. So if I were to plug in a x for the derivative I would get the best approximation of the density of the line at x as well as the rate at which the mass is changing with respect to x at that point?
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The best linear approximation of $y$ near a point $(x_0,y_0)$ is given by $y \approx y_0 + y'(x_0)(x-x_0)$. It's not just the value of the derivative. The density per unit length at $x_0$ is technically exactly given by $m'(x_0)$, though. – Ian Sep 30 '21 at 23:58
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Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Sep 30 '21 at 23:58
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Your description of $m(x)$ is a bit vague when you say " the amount of mass at a certain length of the line." Perhaps you have in mind a cumulative distribution function, so that $m(x) = 0$ for $x$ at one end of "the line" and $m(x)$ is the total mass present when $x$ is at the other end of the line. In that case the derivative $m'(x)$ would be a density function which describes the concentration of mass along points of the line. – hardmath Oct 01 '21 at 00:19