c Dr Oksana Shatalov, Spring 2012
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3.8: Higher Derivatives
The derivative of a differentiable function f is also a function and it may have a derivative of its own:
(f 0 )0 = f 00
second derivative
d 0 d
d
00
f (x) =
f (x) =
f (x)
dx
dx dx
Alternative Notation:
If y = f (x) then
y 00 = f 00 (x) =
d2 y
= D2 f (x).
dx2
Similarly, the third derivative f 000 = (f 00 )0 or
d3 y
d d2 y
000
000
=
= D3 f (x).
y = f (x) =
dx dx2
dx3
In general, the nth derivative of y = f (x) is denoted by f (n) (x):
d dn−1 y
(n)
(n)
= Dn f (x).
y = f (x) =
dx dxn−1
EXAMPLE 1. If y = x5 + 3x + 1 find f (n) (x)
CONCLUSION: If p(x) is a polynomial of degree n then, p(k) (x) = 0 for k ≥ n + 1.
EXAMPLE 2. Find the second derivative of f (x) = tan(x3 ).
c Dr Oksana Shatalov, Spring 2012
EXAMPLE 3. Find D2016 sin x.
EXAMPLE 4. If f (x) =
1
x
find a general formula for its nth derivative.
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c Dr Oksana Shatalov, Spring 2012
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Acceleration: If s(t) is the position of an object then the acceleration of the object is the first
derivative of the velocity (consequently, the acceleration is the second derivative of the position function.)
a(t) = v 0 (t) = s00 (t).
EXAMPLE 5. If s(t) = t3 − 92 t2 − 30t + 12 is the position of a moving object at time t (where s(t) is
measured in feet and t is measured in seconds) find the acceleration at the times when the velocity is zero.
EXAMPLE 6. Sketch the curve traced by r(t) = hcos t, sin ti and plot the position, tangent and accelerπ
ation vectors at t = .
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