3.1 – Derivatives of
Polynomials and
Exponential Functions
1
Differentiation Rules
Let c be a constant and n be any real
number.
d
1.
(c )  0
dx
d
2.
( x)  1
dx
2
The Power Rule
Let c be a constant and n be any real
number.
d n
n 1
3.
( x )  nx
dx
d
d n
n
n 1
4a.
(cx )  c  x   c  nx
dx
dx
3
Example
Differentiate each function.
1. f ( x)  57
2. g ( x)  3 x
5
1 3/ 7
3. H (t )  t
5
10
4. h( x) 
5 9
x
4
The Sum/Difference Rules
Let c be a constant and let f(x) and g(x)
be differentiable functions.
d
d
4b.
cf ( x)  c f ( x)
dx
dx
d
d
d
5.
 f ( x)  g ( x)   f ( x)  g ( x)
dx
dx
dx
5
The Definition of e
e is the number such that
e 1
lim
1
h 0
h
h
d x
x
6.
(e )  e
dx
6
Example
Differentiate each function.
1 4
2
1. f (m)  m  3m  m  2
6
2. g ( x)  x


x 1
x 2 x
3. y 
x
2
4. h(t )  t  2 t  3e
3
2
3
t
7
Example
Determine f , f , f .
f  x  e  x
x
3
8
Example
The equation of motion of a particle is
s(t) = 2t3 – 7t2 + 4t + 1, where s is in meters and
t is in seconds.
(a) Find the velocity and acceleration as
functions of t.
(b) Find the acceleration after 1 second.
(c) Find when the velocity is 0 m/s.
(d) Graph the position, velocity, and
acceleration functions on the same screen.
9