Section 3.8: Higher Derivatives
Definition: The second derivative of a function f (x) is the derivative of the first derivative.
That is,
d 0
d df
d2 f
00
f (x) =
[f (x)] =
= 2.
dx
dx dx
dx
Similarly, the third derivative of f (x) is the derivative of the second derivative,
d d2 f
d3 f
d 00
000
[f (x)] =
=
f (x) =
.
dx
dx dx2
dx3
This process can be continued to define other higher derivatives. In general, the nth derivative of f is denoted by f (n) (x).
Example: Find all higher derivatives of f (x) = x4 − 3x3 + 5x2 + 16x − 2.
Example: Find all higher derivatives of f (x) =
1
1
.
x
Example: If f (x) = cos(2x), find f (61) (x).
Example: The position of a particle at time t is s(t) = 2t3 − 7t2 + 4t + 1, where s is in meters
and t is in seconds.
(a) Find the velocity and acceleration of the particle at time t.
(b) Find the acceleration at the instants when the velocity is zero.
2
~
Example: Consider the vector function R(t)
= hcos t, sin ti.
~
(a) Sketch the graph of R(t).
~ 0 (t) and R
~ 00 (t).
(b) Find R
~
~ 0 (t), and the acceleration vector
(c) Sketch the position vector R(t),
the tangent vector R
~ 00 (t) at t = π/4.
R
3
Example: Find y 00 by implicit differentiation.
x2 + 6xy + y 2 = 8
4