1. Sec 2.8: The Derivative as a Function
Def:
The derivative of a function ƒ at a point x0
DEFINITION
If ƒ’(a) exists, we say that ƒ is differentiable at a.
(has a derivative at a)
ƒ is differentiable at 2

2. Sec 2.8: The Derivative as a Function
DEFINITION
Notations
3) is ƒ differentiable at 3 ?

3. Sec 2.8: The Derivative as a Function
DEFINITION
Right-hand derivative at a
exist
DEFINITION
Left-hand derivative at a
exist
2) Find the right-hand derivative at 0
3) Find the left-hand derivative at 0

4. Sec 2.8: The Derivative as a Function
DEFINITION
DEFINITION
If ƒ’(a) exists, we say that ƒ is differentiable at a.
(has a derivative at a)
A function f is differentiable on an open interval (a, b) if it is differentiable at every number in the interval.
Example:
ƒ is differentiable on (3, 4)
Example:
ƒ is differentiable on
Example:
ƒ is not differentiable on

5. Slopes (negative, positive, zero)
Slope is +1
Slope is -1
Slope is zero
Slope is negative
Slope is positive

6. Sec 2.8: The Derivative as a Function
Find:
When:
Positive or Negative:
When it is positive:

7. Sec 2.8: The Derivative as a Function

8. Sec 2.8: The Derivative as a Function
Sketch the Graph of the derivative of the function

9. Sec 2.8: The Derivative as a Function

10. Sec 2.8: The Derivative as a Function

11. Theorem: Sec 2.8: The Derivative as a Function Proof:
Differentiability
Continuity
Both continuity and differentiability are properties for a function. The following theorem shows how these properties are related.
Theorem:
Differentiable at
Continuous at
Proof:

12. Theorem: Sec 2.8: The Derivative as a Function Remark: Remark: Remark:
Differentiable at
Continuous at
Remark:
f cont. at a
f diff. at a
Remark:
f discont. at a
f not diff. at a
Remark:
f not diff. at a
f discont. at a

13. Sec 2.8: The Derivative as a Function
f cont. at a
f diff. at a
f discont. at a
f not diff. at a
f not diff. at a
f discont. at a
Example:

14. Sec 2.8: The Derivative as a Function
HOW CAN A FUNCTION FAIL TO BE DIFFERENTIABLE?
corner
discontinuity
vertical tangent
kink
corner
discontinuity
discontinuity
vertical tangent,
oscillates

15. Sec 2.8: The Derivative as a Function

16. Sec 2.8: The Derivative as a Function
Differentiable at
exists
Continuous at

17. Sec 2.8: The Derivative as a Function
TERM-122 Exam-2

18. Sec 2.8: The Derivative as a Function
Higher Derivative
Note:
Example:
velocity
acceleration
jerk

19. Sec 2.8: The Derivative as a Function

20. Sec 2.8: The Derivative as a Function