HIGHER ORDER DERIVATIVES
(Velocity and Acceleration)
Since the derivative of a function, f, is itself a function, f ’, its
derivative can be taken, (f ’)’. The result is a function called
the second derivative.
If y = f(x), then:
f '( x ) 
dy
dx
first derivative
ex.
Example 
f ( x )  ( x 2  3)5
b)
y x
x 1
second derivative
If 𝑦 = 2𝑥 5 , then 𝑦 ′ = 10𝑥 4 and 𝑦 ′ ′ = 40𝑥 3 .
Determine f ''( x ) for each of the following:
a)
2
dy
d y
f ''( x )  d    2
dx  dx  dx
Consider an object that moves along a straight line with an origin of reference as well as
positive and negative directions (ie) a number line.
Motion on a Straight Line
position:
𝑠(𝑡)
 the position of the object on the line relative
to the origin as a function of time
velocity:
v(t )  s '(t )
or v  ds
dt
s
 rate of change of s(t)
with respect to time
t
acceleration:
a(t )  v '(t )
or a  dv
dt
 s ''(t )
v
 the rate of change of 𝑣(𝑡)
with respect to time
t
POINTS to NOTE:
1. Positive velocity:
 v(t) > 0
 the
object
is moving
Note: the speed
of the
object
is v(t ) in
. a positive direction (right or up)
2. Negative velocity:
 v(t) < 0
 the object is moving in a negative direction (left or down)
3. Zero velocity:
 v(t) = 0
 the object is not moving (may be changing directions)
4. Positive acceleration:
 a(t) > 0
 the velocity is ______________________________
5. Negative acceleration:  a(t) < 0
 the velocity is ______________________________
6. Zero acceleration:
 a(t) = 0
 the velocity is ______________________________
7. An object is accelerating (speeding up) when its velocity and acceleration have the
same signs (ie) both positive or both negative.
8. An object is decelerating (slowing down) when its velocity and acceleration have
opposite signs (ie) one is positive and the other is negative.
Example 
a)
An object is moving along a straight line. Its position, s(t), is given by
the graph shown.
When is the object moving to the right?
To the left?
When is it at rest?
b)
Sketch the object’s motion on a position diagram.
Example 
The position of an object moving on a line is given by s(t) = 4t2 – t3, t  0,
where s is in metres and t is in seconds.
a)
Determine whether the object is moving in a positive or negative direction
at t = 2s and t = 3s.
b)
Determine the acceleration of the object at t = 3s. Is the object speeding up or
slowing down at this time?
Homework: p.127–129 #1–4, 6, 9, 12