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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.4 • Differentiating the Product and the Quotient of Two Functions; Higher-Order Derivatives 209

Leibniz notation also can be used for higher-order derivatives. Table 3 summarizes
the notation for higher-order derivatives.

TABLE 3
Prime Notation Leibniz Notation
dy d
First Derivative y ′ f (x) f (x)
′
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dx dx
2
d y d 2
Second Derivative y ′′ f (x) f (x)
′′
dx 2 dx 2
3
d y d 3
Third Derivative y ′′′ f (x) f (x)
′′′
dx 3 dx 3
4
d y d 4
Fourth Derivative y (4) f (4) (x) f (x)
dx 4 dx 4
.
.
.
n
d y d n
(n)
(n)
nth Derivative y f (x) f (x)
dx n dx n
EXAMPLE 7 Finding Higher-Order Derivatives of a Power Function
3
Find the second, third, and fourth derivatives of y = 2x .
Solution
Use the Simple Power Rule and the Constant Multiple Rule to find each derivative. The
first derivative is
d d
′ 3 3 2 2
y = (2x ) = 2 · x = 2 · 3x = 6x
dx dx
The next three derivatives are
d 2 d d
′′ 3 2 2
y = 2 (2x ) = (6x ) = 6 · x = 6 · 2x = 12x
dx dx dx
d 3 d
′′′ 3
y = 3 (2x ) = (12x) = 12
dx dx
d 4 3 d
(4)
y = (2x ) = 12 = 0
dx 4 dx
All derivatives of this function f of order 4 or more equal 0. This result can be
generalized.
For a power function f of degree n, where n is a positive integer,

f (x) = x n
f (x) = nx n − 1
′
NOTE If n > 1 is an integer, the product f (x) = n(n − 1)x n − 2
′′
n · (n − 1) · (n − 2) · . . . · 3 · 2 · 1 .
is often written n! and is . .
read, “n factorial.” f (n) (x) = n(n − 1)(n − 2) · . . . · 3 · 2 · 1
The factorial symbol !
means 0! = 1, 1! = 1, and n
n! = 1 · 2 · 3 · . . . · (n − 1) · n, The nth-order derivative of f (x) = x is a constant, so all derivatives of order
where n > 1. greater than n equal 0.

It follows from this discussion that the nth derivative of a polynomial of degree n
is a constant and that all derivatives of order n + 1 and higher equal 0.
NOW WORK Problem 41.

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