Basic Differentiation Rules and Rates of Change
The Constant Rule
The derivative of a constant function is 0. In other
words, if c is a real number, then
d
c  0
dx
The Power Rule:
 
If n is a rational
number, then the function
n
d n
f ( x)  x is differentiable and
x  nx n 1
dx
(be careful at x=0)
The Constant Multiple Rule:
(if f is a differentiable function.)
d
[cf ( x)]  cf ' ( x)
dx
The Sum and Difference Rules
If f and g are differentiable, then their sum and
difference are also differentiable. Furthermore,
 
d x
e  ex
dx
d
 f ( x)  g ( x)  f ' ( x)  g ' ( x)
dx
Trig and Exponential Derivatives
d
sin x  cos x
dx
d
cos x   sin x
dx
d x
x

e  e
dx
A function s that gives position
(relative to the origin) of an
object as a funciton of time t is
called a position funciton.
Average velocity = distance / time
For a position function s(t), the average
velocity over the time interval [t,t+h] is
s (t  h)  s (t )
h
To find instantaneous velocity,
what should we do to the length of
the time interval?
Make it approach zero
Example:
• #94
• A ball is thrown straight down from the top
of a 220-foot building with an initial velocity
of – 22 feet per second. What is its
velocity after 3 seconds? What is its
velocity after falling 108 feet?