Math 151 notes Section 3.1
Derivatives
Definition: The derivative of f ( x ) at x  a is lim
f (a  h)  f (a )
h 0
if this limit exists.
h
If the above limit exists, f ( x ) is called differentiable at x  a and the derivative is called
f ' ( a ) , read as ‘ f prime at a ’
f (a  h)  f (a )
The expression
is called the difference quotient and represents the slope of the
h
secant line connecting the two points, ( a , f ( a )) and ( a  h , f ( a  h )) .
Other commonly used notation is f ' ( a )  lim
f ( x)  f (a )
xa
x a
.
The definition says that f ' ( a ) is the limit of the slopes of the secant lines through ( a , f (a )) .
f ' ( a ) is the slope of the line tangent to the graph of f ( x ) at ( a , f (a )) .
Example: For f ( x )  x
2
find f ' ( 3 ) . Find the equation of the tangent line at (3,9) and sketch the
function and its tangent line.
In general, the equation of the tangent line is y  f ' ( a )( x  a )  f ( a )
The derivative function is f ' ( x )  lim
h 0
f ( x  h)  f ( x)
h
Examples: Find the equation of the tangent line to to f at (a, f(a)):
1.
f (x)  x
2.
f ( x) 
3.
f ( x) 
a5
2
x
1
x
a4
a2
If f ( x ) is differenti able on ( c , d ), then f ( x  a ) is differenti able
The shift rule:
on ( c  a , d  a ) and has derivative
f ( x )  x and g ( x )  ( x  1) .
2
Example: Sketch
2
f ' ( x  a ).
Sketch the
tangent line to
f at a  2 and to g at a  3 .
Sketch any differentiable function and a tangent line. Shift to the right or left and the tangent line shifts
the same way.
Examples: Find the derivative of each function.
1.
f ( x )  ( x  5)
2.
f ( x) 
x2
3.
f ( x) 
1
2
x3
Fact: If f ( x ) is differentiable at a , then it is continuous at a .
Contrapositive: If f ( x ) is not continuous at a , then it is not differentiable at a .
Show that if lim
x a
f ( x)  f (a )
xa
 f ' ( a ) exists, then lim ( f ( x )  f ( a ))  0
xa
The converse is false: Examples:
1.
2 x  3
f ( x)  
 5x
x 1
x 1
Sketch the graph and show f ( x ) is continuous but is not differentiable at 1.
2.
f (x)  x
1 3
is continuous at 0 but is not differentiable at 0.
The Linear Property:
For any two constants
f ( x ) and g ( x ),
 Af
A and B and any two differenti able functions
 Bg  '  Af ' Bg '.
Use the derivatives we found and apply the linear property to find each derivative.
1.
f ( x)  5 x  7 x  4
2.
g ( x)  6 x 
2
8
x
3.
h ( x )  3 ( x  4 )  16
2
x  2  9 x  12