1. Derivative as a function Math 1231: Single-Variable Calculus

2. Derivative as a function We can regard f’(x) as a new function, called the derivative of f.

3. Examples

4. Other Notations

5. Continuity v.s. Differentiability If f is differentiable at a, then f is continuous at a; Reversely, if a function is continuous at a, then f is NOT necessarily differentiable at a. In other words, Differentiability is stronger than continuity. Example: f(x) = |x| is continuous at x = 0, but fail to be differentiable at x = 0.

6. Non-differentiable function

7. Higher Derivatives

8. Examples f(x)  f’(x)  f’’(x)  f’’’(x) position  velocity  acceleration  jerk