Relationship between Linear Regression and derivative of a function?

Question

If I am given a nice function (not necessarily linear) $f(x)$, then I can find the best linear fit $L(x)$ for this function (using linear regression). Also I can find the derivative of the function at each point which is $f'(x)$ (which is also the best linear approx near $x$). Now am I trying to see the connection between derivative of $f(x)$ and its best linear fit $L(x)$. My general intuition is that $L(x) = $ some sort of average of $f'(x)$. Are there any theorem that explores/answer this questions.