1. Sec 3.2: The Derivative as a Function If ƒ’(a) exists, we say that ƒ is differentiable at a. (has a derivative at a) DEFINITION

2. Sec 3.2: The Derivative as a Function Calculating Derivatives from the Definition Notations ƒ is differentiable at 2 If ƒ’(a) exists, we say that ƒ is differentiable at a. (has a derivative at a) DEFINITION

3. Sec 3.2: The Derivative as a Function Calculating Derivatives from the Definition ƒ is differentiable at 3 If ƒ’(a) exists, we say that ƒ is differentiable at a. (has a derivative at a) DEFINITION Differentiate

4. Sec 3.2: The Derivative as a Function Right-hand derivative at a DEFINITION exist Left-hand derivative at a DEFINITION exist Find the right-hand derivative at 0 and the left-hand derivative at 0

5. Definition: A function f is differentiable on an open interval (a, b) if it is differentiable at every number in the interval. Definition: A function f is differentiable on an open interval (a, b) if it is differentiable at every number in the interval. Sec 3.2: The Derivative as a Function If ƒ’(a) exists, we say that ƒ is differentiable at a. (has a derivative at a) DEFINITION

6. Slopes : 0 + - Sec 3.2: The Derivative as a Function

9. Sketch the Graph of the derivative of the function

10. Sec 3.2: The Derivative as a Function

13. 2 properties continuity differentiability Proof: Sec 3.2: The Derivative as a Function Continuous at Differentiable at

14. 2 properties continuity differentiability Proof: Remark: f cont. at af diff. at a Remark: f discont. at af not diff. at a Remark: f discont. at a f not diff. at a Sec 3.2: The Derivative as a Function

15. Example: f cont. at af diff. at a f discont. at af not diff. at a f discont. at a f not diff. at a Sec 3.2: The Derivative as a Function

16. Example: f cont. at af diff. at a f discont. at af not diff. at a f discont. at a f not diff. at a Sec 3.2: The Derivative as a Function

17. TERM-121 Exam-2

18. HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE? Sec 3.2: The Derivative as a Function corner cusp vertical tangent, discontinuity oscillates

19. Sec 3.2: The Derivative as a Function

20. exists Continuous at Differentiable at

21. Sec 3.3: Differentiation Rules TERM-121 Exam-2

22. Sec 3.3: Differentiation Rules TERM-122 Exam-2

23. Sec 3.3: Differentiation Rules