Download 3.1: Derivative of a Function

3.1: Derivative of a Function
OBJECTIVES:
•T O F I N D T H E D E R I V A T I V E O F A F U N C T I O N
•T O U S E T H E D E R I V A T I V E T O W R I T E
EQUATIONS OF A TANGENT AND NORMAL
LINE
•T O G R A P H T H E D E R I V A T I V E F U N C T I O N
DEFINTION
The derivative of the function f with respect to the
variable x is the function f ‘ whose value at x is
f ( x  h)  f ( x )
f ' ( x)  lim
h 0
h
PROVIDED THE LIMIT EXISTS
Look familiar???
The derivative is….





The slope of the curve at a point
The slope of the tangent line to the curve at a point
Instantaneous rate of change
Instantaneous velocity if the function f represents
position
A function with its own domain and range (may be
different from domain of f)
NOTATIONS FOR DERIVATIVE OF y = f(x)
 Pay attention to function name and variables used
f ’(x)
“f prime of x”
y’
“y prime”
dy
dx
df
dx
“dee y dee x; the derivative
of y with respect to x”
d
f (x)
dx
“dee f dee x; the derivative
of f with respect to x”
“d dx of f at x” or “the
derivative of f at x”
Given f(x)=x3 +2x. Find f ’ (x) and f ’ (2).
Find f ‘ (x):



f ( x  h)  f ( x )
x  h   2 x  h   x 3  2 x
lim
 lim
h 0
h 0
h
h
3
3 x 2 h  3 xh 2  h 3  2h
 lim
 3x 2  2
h 0
h
Answer: f ‘(x) = 3x2 +2
Then, substitute 2 into the derivative function:
f’(2)= 3(2)2 +2 = 14

d
Find dx
 x .
Write the equation for the tangent
and normal line at x = 1.
Remember, the derivative gives you the slope of the curve at any point, as
well as, the slope of the tangent line.
Conjugate Method
d
dx
 
xh  x
x  lim
 lim
h 0
h 0
h

1
1

xh  x
2 x

At x = 1, dy/dx = ½.
Tangent Line: y = (1/2)x +(1/2)
Normal Line: y = -2x +3
QUESTION: Where is f(x) defined but f ‘ (x) not defined?
Remember, the derivative is a function with its own domain
and range, which may differ from f(x).
ALTERNATE DEFINTION
 Useful when trying to find derivative at a point.
f ( x)  f (a)
f ' (a)  lim
xa
xa
PROVIDED THE LIMIT EXISTS
Given f(x)=2x3+3. Find f’(x) and f’(6).
Find
d  1 
 2
dx  x 
Relationship Between Graph of f and f ‘
 Derivative at a point = slope
 We can get a good idea of the graph of f ’ by
estimating the slopes at various points along the
graph of f
 Points on the derivative graph will be (x, f ’ (x))
Conclusions…
 f is increasing where f ’ is positive
 f is decreasing where f ‘ is negative
 x-intercept of the graph of f ‘ is where there is
a horizontal tangent line to the graph of f
(When graphing derivative, helps to find
these points first)
Graph the Derivative
http://webspace.ship.edu/msrenault/GeoGebraCalcul
us/derivative_try_to_graph.html
Graphing the Derivative from Data
2 techniques:
1. Create scatter plot of data, use regression equation
to get a graph of the derivative
2. Scatter plot of derivative points by numerically
computing slopes between successive points
#11 on worksheet
One Sided Derivatives
Right-hand Derivative
f ( x  h)  f ( x )
lim
h 0
h
Left-hand Derivative
f ( x  h)  f ( x )
lim
h 0
h
•In order for the derivative at x=a to exist, the left hand derivative has to
equal the right hand derivative
•Note a function can be continuous at a point but not differentiable!!
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
f ( a  h)  f ( a )
lim
h 0
h
f (b  h)  f (b)
lim
h 0
h
exist at the endpoints.
Does
x3 , x  0
y
2 x  1, x  0
have a derivative at x=0?