1. Section 3.2
Differentiability

2. How f’(a) Might Fail to Exist
A corner, where the one-sided derivatives differ

3. How f’(a) Might Fail to Exist
A cusp, where the slopes of the secant lines approach  from one side and - from the other (an extreme case of a corner):

4. How f’(a) Might Fail to Exist
A vertical tangent, where the slopes of the secant lines approach either  or - from both sides.

5. How f’(a) Might Fail to Exist
A discontinuity, which will cause one or both of the one-sided derivatives to be nonexistent.

6. Example 1
Find all the points in the domain where f is not differentiable.

7. Example 2
Compute the numerical derivative.

8. Example 3
Compute the numerical derivative.

9. Example 4
Let f(x) = lnx. Use NDERIV to graph y = f’(x). Can you guess what function f’(x) is by analyzing its graph?

10. Differentiability Implies Continuity
Theorem 1: Differentiability Implies Continuity:
If f has a derivative at x = a, then f is continuous at x = a.

11. Intermediate Value Theorem for Derivatives
Theorem 2: Intermediate Value Theorem for Derivatives:
If a and b are any two points in an interval on which f is differentiable, then f’ takes on every value between f’(a) and f’(b).