3.1 definition of the derivative & graphing the derivative
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3.1 Definition of the Derivative & Graphing the Derivative
These are all ways to denote the derivative
y y prime
dy
dx
dy dx or the derivative of y with respect to x
df
dxdy dx or the derivative of f with respect to x
( )df x
dxd dx of f at x or the derivative of f at x
f
( )f x
f prime or the derivative of f
f prime of x or the derivative of f(x)
The derivative of a function f with respect to the variable x is the function f’ whose value at x is
0 0
( ) ( ) ( ) ( )( ) lim lim
h h
f x h f x f x h f xf x
x h x h
Provided the limit exists
, ( )x f x , ( )x h f x h
x x h
h
( ) 3 2f x x
Find the equation of the derivative using the definition of the derivative
0
( ) ( )( ) lim
h
f x h f xf x
h
( ) 3 2 3 3 2f x h x h x h
0
( ) ( )( ) lim
h
f x h f xf x
h
0
3 3 2 3 2limh
x h x
h
0
3 3 2 3 2limh
x h x
h
0
3limh
h
h
0lim3 3h
2( ) 2f x x x
Find the equation of the derivative using the definition of the derivative
0
( ) ( )( ) lim
h
f x h f xf x
h
2( ) 2f x h x h x h
2( )( )x h x h x h
2 22 x xh xh h x h
2 22 2x xh h x h
2 22 4 2x xh h x h
2( ) 2f x x x
Find the equation of the derivative using the definition of the derivative
0
( ) ( )( ) lim
h
f x h f xf x
h
2 2( ) 2 4 2f x h x xh h x h
0
( ) ( )( ) lim
h
f x h f xf x
h
2 2 2
0
2 4 2 2limh
x xh h x h x x
h
2 2 2
0
2 4 2 2limh
x xh h x h x x
h
2
0
4 2limh
xh h h
h
0
4 2 1limh
h x h
h
0lim4 2 1 4 2 0 1 4 1h
x h x x
2( ) 2f x x x
Find the equation of the tangent line to f(x) at x=2
0
( ) ( )( ) lim
h
f x h f xf x
h
4 1f x x
2(2) 2 2 2 6f
2 4 2 1 7f
1 1y y m x x
6 7 2y x
7 8y x
Graph of Derivative
Graph the derivative
Graph the derivative
m=17
m=10
m=0
m= -5
m= -6 m= 0m= 8
m= 15