righting an old wrong

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 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.

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