When Does a Function NOT have a Derivative at a Point?

A function has a derivative at a point x_{0} if the slopes of the secant lines through P(x_{0}, f(x_{0})) and a nearby point Q on the graph approach a limit as Q approaches P. Whenever the secant fail to take a limiting position or become vertical as Q approaches P, the derivative does not exist. A function, whose graph is other wise “smooth” will fail to have a derivative at a point where the graph has:

  1. a Corner, where the one-sided derivatives differ
  2. A Cusp where the slope of PQ approaches \infty from one side and -\infty from the other.
  3. A Vertical Tangent, when the slope of PQ approaches \infty from both sides or approaches -\infty from both sides.
  4. A Discontinuity.

More later,

Nalin Pithwa

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