A function has a derivative at a point if the slopes of the secant lines through 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:

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

More later,

Nalin Pithwa