## 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

This site uses Akismet to reduce spam. Learn how your comment data is processed.