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Looking at reciprocal functions, they have 2 “turning” points, where the higher derivative values near vertical asymptotes turn into the generally lower ones approaching the end behavior asymptote, for example, taking $$y=\frac1x$$ the points are $(1,1)$ and $(-1,-1)$. It appears that these points are where the derivative is equal to $1$ or $-1$.

Looking at exponential and logarithmic functions, they also have these points where the value of the derivatives begin to sharply decrease or increase.

I’m wondering whether these points have a specific name?

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  • $\begingroup$ Perhaps you are looking for points of greatest curvature? $\endgroup$
    – Théophile
    Nov 16, 2018 at 18:49
  • $\begingroup$ What, exactly, is your definition of "the higher derivative values near vertical asymptotes" and "the generally lower ones approaching the end behavior asymptote"? Those seem like very vague concepts to me. $\endgroup$
    – Arthur
    Nov 16, 2018 at 19:01
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    $\begingroup$ For example, y=1/x, the derivative is $-1/x^2$. In the domain (-1,0) and (0,1), the absolute value of the derivative is greater than one, but past those points, the absolute value of the derivative is always less than 1. $\endgroup$
    – H Huang
    Nov 16, 2018 at 21:18

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