CK12 Calculus - Single Variable

Doesnot exist

then the derivativef(x) also fails to existas x → x 0. The followingexamplesshowfour caseswhere
the derivativefails to exist.

1. At acorner. For examplef(x) = |x|, wherethe derivativeon both sidesofx= 0 differ (Figure4).


2. At acusp. For examplef(x) =x2/3, wherethe slopesof the secantlines approach+∞ on the right and -∞
   on the left (Figure5).


3. Averticaltangent. For examplef(x) =x1/3, wherethe slopesof the secantlines approach+∞ on the right
   and -∞ on the left (Figure6).


4. Ajumpdiscontinuity. For example,the step function(Figure7)

wherethe limit from the left is -2 and the limit from the right is 2.