Differential Equations For Dummies
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There are three situations where a derivative fails to exist. The derivative of a function at a given point is the slope of the tangent line at that point. So, if you can’t draw a tangent line, there’s no derivative — that happens in cases 1 and 2 below. In case 3, there’s a tangent line, but its slope and the derivative are undefined.

The three situations are shown in the following list.

  1. When there’s no tangent line and thus no derivative at any of the three types of discontinuity:

    • A removable discontinuity — that’s a fancy term for a hole — like the holes in functions r and s in the above figure.

    • An infinite discontinuity like at x = 3 on function p in the above figure.

    • A jump discontinuity like at x = 3 on function q in the above figure.

      Continuity is, therefore, a necessary condition for differentiability. It’s not, however, a sufficient condition as the next two cases show. Dig that logician-speak.

  2. When there’s no tangent line and thus no derivative at a sharp corner on a function. See function f in the above figure.

  3. Where a function has a vertical inflection point. In this case, the slope is undefined and thus the derivative fails to exist. See function g in the above figure.

About This Article

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About the book author:

Steven Holzner was an award-winning author of more than 130 books, of which more than 2 million copies have been sold. His books have been translated into 23 languages. He served on the Physics faculty at Cornell University for more than a decade, teaching both Physics 101 and Physics 102. Holzner received his doctorate in physics from Cornell and performed his undergraduate work at Massachusetts Institute of Technology, where he also served as a faculty member.

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