1. 3.2 Differentiability

2. Differentiability
A function is differentiable at point c if and only if the derivative from the left of c equals the derivative from the right of c.
AND
if c is in the domain of f’.

3. Differentiability Find the derivative of at x = 0.
f is not differentiable at 0.

4. DIFFERENTIABILITY
A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth.
No “sudden change” in slope.

5. DIFFERENTIABILITY Derivatives will fail to exist at: corner
vertical tangent
cusp
any discontinuity

6. Using the calculator
The numerical derivative of f at a point a can be found using NDER on the calculator.
Syntax: NDER (f(x), a)
Note: The calculator uses h = to compute the numerical derivative, so it is a close approximation to the actual derivative.
Example: Compute NDER of f(x) = x3 at x = 2.

7. differentiability THEOREM: Differentiability implies continuity.
If f has a derivative at x = a, then f is continuous at x = a.
Differentiability implies continuity.