Finite notes – 8.5
Higher Order Derivatives
The second derivative, or second order derivative, is the derivative of the derivative of a function. The
derivative of the function f(x) may be denoted by
, and its double (or "second") derivative is
denoted by
. This is read as "f double prime of x," or "The second derivative of f(x)." Because the
derivative of function f is defined as a function representing the slope of function f, the double derivative
is the function representing the slope of the first derivative function.
Furthermore, the third derivative is the derivative of the derivative of the derivative of a function, which
can be represented by
. This is read as "f triple prime of x", or "The third derivative of f(x)". This
can continue as long as the resulting derivative is itself differentiable, with the fourth derivative, the fifth
derivative, and so on. Any derivative beyond the first derivative can be referred to as a higher order
derivative.
Let f(x) be a function in terms of x. The following are notations for higher order derivatives.
2nd Derivative 3rd Derivative 4th Derivative nth Derivative
Notes
f(4)(x)
f(n)(x)
Probably the most common notation.
Leibniz notation.
Another form of Leibniz notation.
D2f
D3f
D4f
Dnf
Euler's notation.
Warning: You should not write fn(x) to indicate the nth derivative, as this is easily confused with the
quantity f(x) all raised to the nth power.
The Leibniz notation, which is useful because of its precision, follows from
.
Newton's dot notation extends to the second derivative, , but typically no further in the applications
where this notation is common.
Finite Math – Section 8.5 - page 2
Example 1:
Find the third derivative of
respect to x.
with
Repeatedly apply the Power Rule to find the derivatives.



Example: Find f ''' ( x ) and f ''' (2) of f(x)=
x-2
.
4x
Distance, Velocity, and Acceleration
If a function, s(t), represents distance at time t, then s’(t), the 1st derivative, represents the velocity,
and s’’(t), the 2nd derivative of s(t), represents acceleration. The 3rd derivative, s’’’(t) represents jerk.
Remember to connect derivatives with “rate of change” of a quantity.
Finite – Section 8.5 page 3
EXAMPLE
Each of the following three “stories,” labeled a, b, and c, matches one of the velocity
graphs, labeled (i), (ii), and (iii). For each story, choose the most appropriate graph.
a.
I left my home and drove to meet a friend, but I got stopped for a speeding ticket.
Afterward I drove on more slowly. _______
b. I started driving but then stopped to look at the map. Realizing that I was going
the wrong way, I drove back the other way. _______
c. After driving for a while I got into some stop-and-go driving. Once past the tie-up I
could speed up again. _______
EXAMPLE
A rocket can rise to a height of h(t )  t 3  0.5t 2 feet in t seconds. Find its velocity and
acceleration 10 seconds after it is launched.
Four interpretations for the derivative:
1. instantaneous rate of change
2. slope of tangent
3. marginals
4. velocity