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

216 Chapter 2 • The Derivative and Its Properties

97. Current Density in a Wire The current density J in a wire 110. If f and g are differentiable functions with f 6= −g, find the
is a measure of how much an electrical current is compressed f g
as it flows through a wire and is modeled by the derivative of f + g .
I
function J(A) = , where I is the current (in amperes) and A 2x
A CAS 111. f (x) = .
is the cross-sectional area of the wire. In practice, current x + 1
′
density, rather than merely current, is often important. For (a) Use technology to find f (x).
example, superconductors lose their superconductivity if the (b) Simplify f to a single fraction using either algebra or a
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′
current density is too high. CAS.
(a) As current flows through a wire, it heats the wire, causing it (c) Use technology to find f (5) (x).
to expand in area A. If a constant current is maintained in a Hint: Your CAS may have a method for finding
cylindrical wire, find the rate of change of the current higher-order derivatives without finding other derivatives
density J with respect to the radius r of the wire. first.
(b) Interpret the sign of the answer found in (a).
Challenge Problems
(c) Find the rate of change of current density with respect to the
radius r when a current of 2.5 amps flows through a wire of 112. Suppose f and g have derivatives up to the fourth order. Find
radius r = 0.50 mm. the first four derivatives of the product f g and simplify the
98. Derivative of a Reciprocal, Function Prove that if a answers. In particular, show that the fourth derivative is
′
d 1 g (x) d 4
function g is differentiable, then = − , ( f g) = f (4) g + 4 f (3) (1) + 6 f (2) (2) + 4 f (1) (3) + f g (4)
g
g
g
dx g(x) [g(x)] 2 dx 4
provided g(x) 6= 0.
99. Extended Product Rule Show that if f, g, and h are Identify a pattern for the higher-order derivatives of f g.
differentiable functions, then 113. Suppose f 1 (x), . . . , f n (x) are differentiable functions.
d d
[ f (x)g(x)h(x)] = f (x)g(x)h (x) + f (x)g (x)h(x) (a) Find [ f 1 (x) · . . . · f n (x)].
′
′
dx dx
+ f (x)g(x)h(x) d 1
′
From this, deduce that (b) Find .
dx f 1 (x) · . . . · f n (x)
d 3 2 114. Let a, b, c, and d be real numbers. Define
′
[ f (x)] = 3[ f (x)] f (x)
dx
a b
In Problems 100–105, use the Extended Product Rule (Problem 99) c = ad − bc
d
to find y .
′
2
100. y = (x + 1)(x − 1)(x + 5) This is called a 2 × 2 determinant and it arises in the study
of linear equations. Let f 1 (x), f 2 (x), f 3 (x), and f 4 (x) be
2
3
101. y = (x − 1)(x + 5)(x − 1)
differentiable and let
4
3
102. y = (x + 1) 3 103. y = (x + 1) 3
f 1 (x)
f 2 (x)

1 D(x) =
104. y = (3x + 1) 1 + (x −5 + 1) f 3 (x) f 4 (x)
x
Show that

1 1 1
′
′
105. y = 1 − 1 − 2 1 − 3 f (x) f (x) f 1 (x) f 2 (x)
1
2
x x x D (x) = +
′
′
′
f 3 (x) f 4 (x) f (x) f (x)
3
4
106. (Further) Extended Product Rule Write a formula for the
derivative of the product of four differentiable functions. That
115. Let f 0 (x) = x − 1
d
is, find a formula for [ f 1 (x) f 2 (x) f 3 (x) f 4 (x)]. Also find a 1
dx f 1 (x) = 1 +
d x − 1
4
formula for [ f (x)] . 1
dx f 2 (x) = 1 +
1
107. If f and g are differentiable functions, show that 1 +
x − 1
1 1
if F(x) = , then
f (x)g(x) f 3 (x) = 1 + 1
1 +
′
′
f (x) g (x) 1
′
F (x) = −F(x) + 1 +
f (x) g(x) x − 1
provided f (x) 6= 0, g(x) 6= 0. ax + b
1 (a) Write f 1 , f 2 , f 3 , f 4 , and f 5 in the form cx + d .
108. Higher-Order Derivatives If f (x) = , find a formula for
1 − x (b) Using the results from (a), write the sequence of numbers
the nth derivative of f . That is, find f (n) (x). representing the coefficients of x in the numerator,
4
6
x − x + x 2 beginning with f 0 (x) = x − 1.
109. Let f (x) = 4 . Rewrite f in the (c) Write the sequence in (b) as a recursive sequence.
x + 1
4
2
4
6
form (x + 1) f (x) = x − x + x . Now find f (x) without Hint: Look at the sum of consecutive terms.
′
′
′
′
′
′
′
using the quotient rule. (d) Find f , f , f , f , f , and f .
0
4
1
2
5
3
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