The derivative of a function is another function. We look at the interplay between the two. Also, new notations, higher derivatives, and some sweet wigs

1. Section 2.8
The Derivative as a Function
Math 1a
February 13, 2008
Announcements
Office Hours TW 2–4 in SC 323
ALEKS is due Wednesday 2/20
HW on website

2. Outline
Cleanup: Derivatives of some root functions
The derivative function
Worksheet #1
How can a function fail to be differentiable?
Other notations
The second derivative
Worksheet #2

3. Last time: Worksheet problems 3 and 4
Problem
Let f (x) = x 1/3 . Find f (x) and its domain.
Problem
Let f (x) = x 2/3 . Find f (x) and its domain.

4. Last time: Worksheet problems 3 and 4
Problem
Let f (x) = x 1/3 . Find f (x) and its domain.
Answer
1
f (x) = x −2/3 . The domain is all numbers except 0.
3
Problem
Let f (x) = x 2/3 . Find f (x) and its domain.

5. Last time: Worksheet problems 3 and 4
Problem
Let f (x) = x 1/3 . Find f (x) and its domain.
Answer
1
f (x) = x −2/3 . The domain is all numbers except 0.
3
Problem
Let f (x) = x 2/3 . Find f (x) and its domain.
Answer
2
f (x) = x −1/3 . The domain is all numbers except 0.
3

6. Outline
Cleanup: Derivatives of some root functions
The derivative function
Worksheet #1
How can a function fail to be differentiable?
Other notations
The second derivative
Worksheet #2

7. The derivative function
We have snuck this in: If f is a function, we can compute the
derivative f (x) at each point x where f is differentiable, and
come up with another function, the derivative function.
What can we say about this function f ?

8. Worksheet #1

9. Outline
Cleanup: Derivatives of some root functions
The derivative function
Worksheet #1
How can a function fail to be differentiable?
Other notations
The second derivative
Worksheet #2

10. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.

11. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.
Proof.
We have
f (x) − f (a)
lim (f (x) − f (a)) = lim · (x − a)
x→a x→a x −a
f (x) − f (a)
= lim · lim (x − a)
x→a x −a x→a
= f (a) · 0 = 0

12. Differentiability is super-continuity
Theorem
If f is differentiable at a, then f is continuous at a.
Proof.
We have
f (x) − f (a)
lim (f (x) − f (a)) = lim · (x − a)
x→a x→a x −a
f (x) − f (a)
= lim · lim (x − a)
x→a x −a x→a
= f (a) · 0 = 0
Note the proper use of the limit law: if the factors each have a
limit at a, the limit of the product is the product of the limits.

13. How can a function fail to be differentiable?
Kinks
f (x)
x

14. How can a function fail to be differentiable?
Kinks
f (x) f (x)
x x

15. How can a function fail to be differentiable?
Cusps
f (x)
x

16. How can a function fail to be differentiable?
Cusps
f (x) f (x)
x x

17. How can a function fail to be differentiable?
Vertical Tangents
f (x)
x

18. How can a function fail to be differentiable?
Vertical Tangents
f (x) f (x)
x x

19. How can a function fail to be differentiable?
Weird, Wild, Stuff
f (x)
x

20. How can a function fail to be differentiable?
Weird, Wild, Stuff
f (x) f (x)
x x

21. Outline
Cleanup: Derivatives of some root functions
The derivative function
Worksheet #1
How can a function fail to be differentiable?
Other notations
The second derivative
Worksheet #2

22. Notation
Newtonian notation
f (x) y (x) y
Leibnizian notation
dy d df
f (x)
dx dx dx

23. Meet the Mathematician: Isaac Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687

24. Meet the Mathematician: Gottfried Leibniz
German, 1646–1716
Eminent philosopher as
well as mathematician
Contemporarily disgraced
by the calculus priority
dispute

25. Outline
Cleanup: Derivatives of some root functions
The derivative function
Worksheet #1
How can a function fail to be differentiable?
Other notations
The second derivative
Worksheet #2

26. The second derivative
If f is a function, so is f , and we can seek its derivative.
f = (f )
It measures the rate of change of the rate of change!

27. The second derivative
If f is a function, so is f , and we can seek its derivative.
f = (f )
It measures the rate of change of the rate of change!
Leibnizian notation:
d 2y d2 d 2f
f (x)
dx 2 dx 2 dx 2

28. Worksheet #2