From: Zack M. Davis Date: Fri, 17 Jul 2026 18:46:18 +0000 (-0700) Subject: righting an old wrong X-Git-Url: https://zackmdavis.net/blog/source?a=commitdiff_plain;h=b21021d7ccfcf9dab5c8289dab68021f463ca523;p=An_Algorithmic_Lucidity.git righting an old wrong --- diff --git a/content/2011/the-derivative-of-the-natural-logarithm.md b/content/2011/the-derivative-of-the-natural-logarithm.md index 18e22df..ae64c30 100644 --- a/content/2011/the-derivative-of-the-natural-logarithm.md +++ b/content/2011/the-derivative-of-the-natural-logarithm.md @@ -6,7 +6,7 @@ Tags: calculus Most people learn during their study of the differential and integral calculus that the derivative of the natural logarithm ln _x_ is the reciprocal function 1/_x_. Indeed, sometimes the natural logarithm is _defined_ as $$ \int_1^x \frac{1}{t}\,dt$$. However, on observing the graphs of ln _x_ and 1/_x_, the inquisitive seeker of knowledge can hardly fail to notice a disturbing anomaly: -[![y=ln(x)]({static}/images/lnx3.png "y=ln(x)")]({static}/images/lnx3.png) [![y=1/x]({static}/images/reciprocalx2.png "y=1/x")]({static}/images/reciprocalx2.png) +[![y=ln(x)]({static}/images/lnx.png "y=ln(x)")]({static}/images/lnx.png) [![y=1/x]({static}/images/reciprocalx.png "y=1/x")]({static}/images/reciprocalx.png) The natural logarithm is only defined for _positive_ numbers; no part of its graph lies in quadrants II or III. But the reciprocal function is defined for all nonzero numbers. So (one _cannot_ help oneself but wonder) how could the latter be the derivative of the former? If the graph of the natural logarithm isn't _there_ to be differentiated in the left half of the plane, how could its derivative be defined in that region? Some would-be explorers lose all hope or sanity in the face of such bizarre and inexplicable mysteries, but even those brave souls who manage to retain their wits are not guaranteed success: many (who can say but that most?) will die never knowing the answer. But not you, dear reader!—for in this very post, I will share with you the _true secret_ of the derivative of the natural logarithm! Some people may find some of what I am about to say somewhat disturbing, even frightening. But if your love of truth exceeds your fear of the unknown, keep reading, and I will show you the strange world that lies beneath these familiar and seemingly innocent graphs. diff --git a/content/images/lnx.png b/content/images/lnx.png new file mode 100644 index 0000000..d8f3714 Binary files /dev/null and b/content/images/lnx.png differ diff --git a/content/images/lnx3.png b/content/images/lnx3.png deleted file mode 100644 index d8f3714..0000000 Binary files a/content/images/lnx3.png and /dev/null differ diff --git a/content/images/reciprocalx.png b/content/images/reciprocalx.png new file mode 100644 index 0000000..1208591 Binary files /dev/null and b/content/images/reciprocalx.png differ diff --git a/content/images/reciprocalx2.png b/content/images/reciprocalx2.png deleted file mode 100644 index 1208591..0000000 Binary files a/content/images/reciprocalx2.png and /dev/null differ