Lesson 3-1: Derivatives

1
Lesson 3-1 Derivatives

• AP Calculus
• Mrs. Mongold

2
is called the derivative of at .
The derivative of f with respect to x is
3
the derivative of f with respect to x
f prime x
or
y prime
the derivative of y with respect to x
or
dee why dee ecks
the derivative of f with respect to x
or
dee eff dee ecks
the derivative of f of x
dee dee ecks uv eff uv ecks
or
4
Note
dx does not mean d times x !
dy does not mean d times y !
5
Note
(except when it is convenient to think of it as
division.)
(except when it is convenient to think of it as
division.)
6
Note
(except when it is convenient to treat it that
way.)
7
(No Transcript)
8
(No Transcript)
9
A function is differentiable if it has a
derivative everywhere in its domain. It must be
continuous and smooth. Functions on closed
intervals must have one-sided derivatives defined
at the end points.
p
10
Example

• Differentiate f(x)x3

11
Example

• Differentiate f(x)1/x

12
Alternate Definition

• Derivative at a point
• The derivative of a function f at the point xa
  is the limit
• Provided the limit exists

13
Example

• Use Alt. Def. to differentiate f(x) at
  xa

14
Example

• Use Alt. Def. to differentiate f(x) at a2

15
Example

• Graph the derivative of f

16
Example

• Graph the derivative of f

17
Example

• If f(x) x3-x, find a formula for f(x) and
  illustrate by comparing f and f graphs

18
Example

• Graph f from f
• Sketch a graph of a function f that has the
  following properties
• f(0)0
• The graph of f, the derivative of f, below
• F is continuous for all x

19
Example

• Sketch the graph of a continuous function f with
  f(0) -1 and

20
One sided derivatives

• A function y f(x) is differentiable on a closed
  interval a, b if it has a derivative at every
  interior point of the interval, and if the limits
• Exist at the endpoints

21
NOTE

• A function has s two-sided derivative at a point
  if and only if the functions RH LH derivatives
  are defined and equal at that point

22
Example

• Show that the following function has left hand
  and right hand derivates at x0 but no derivative
  at x0

23
Homework

• Day 1 pgs 101-104/1-12
• Day 2 Pgs 101-104/ 13, 14, 16, 17, 18, 22, 26, 28