2.2 Basic Differentiation Rules
and Rates of Change
The Constant Rule
d
c  0
dx
Power Rule
 
d n
x  nxn 1
dx
The Constant Multiple Rule
d
cf ( x)  cf ' ( x)
dx
The Sum and Difference Rule
d
 f ( x )  g ( x )   f ' ( x )  g ' ( x)
dx
Examples
1)
f ( x)  2 x  1
2)
5
f ( x)  x  6 x  x  16
3
3)
3
f ( x)  4 x
2
2
 8x  1
Examples
4)
Does the curve y  x 4 -2 x 2  2 have any horizontal tangents?
If so, where?
5)
Find an equation for the line tangent to the
x2  3
curve y 
.
2x
Derivatives of Sine and Cosine
d
sin x  cos x
dx
d
cos x   sin x
dx
Examples
6)
1
f ( x)   x  cos x
x
7)
f ( x)  3 x  5 sin x
Examples
8)
9)
10)
5
f ( x)  3
2x
f ( x) 
1
x x
43 x  x cos x
f ( x) 
x
Rates of Change
Position funtion :
s (t )
Change in distance s
Average velocity 

Change in time
t
Instantaneous velocity :
Rates of Change
At time t  0, a diver jumps from a diving board that
is 32 feet above the water. The position of the diver
is given by
s (t )  16t 2  16t  32
where s is measured in feet and t is measured in seconds.
1) When does the diver hit the water?
2) What is the diver' s average velocity over the course
of the entire dive?
3) How fast is the diver moving when she hits the water?