Chapter 3 | Derivatives
301
Solution The function g ( x ) = x 3 is the inverse of the function f ( x ) = x 3 . Since g ′( x ) = 1 f ′ ⎛ ⎝ g ( x )
⎞ ⎠ , begin by finding
f ′( x ). Thus,
2
⎛ ⎝ x 3
⎞ ⎠
=3 x 2/3 .
f ′( x ) =3 x 2 and f ′ ⎛
⎞ ⎠ =3
⎝ g ( x )
Finally,
x −2/3 .
g ′( x ) = 1
= 1 3
3 x 2/3
3.43
Find the derivative of g ( x ) = x 5 by applying the inverse function theorem.
From the previous example, we see that we can use the inverse function theorem to extend the power rule to exponents of the form 1 n , where n is a positive integer. This extension will ultimately allow us to differentiate x q , where q is any rational number.
Theorem 3.12: Extending the Power Rule to Rational Exponents The power rule may be extended to rational exponents. That is, if n is a positive integer, then
d dx (1/ n )−1 . Also, if n is a positive integer and m is an arbitrary integer, then ⎞ ⎠ = 1 n x ⎛ ⎝ x 1/ n
(3.20)
⎛ ⎝ x m / n
⎞ ⎠ = m
d dx
(3.21)
( m / n )−1 .
n x
Proof The function g ( x ) = x 1/ n is the inverse of the function f ( x ) = x n . Since g ′( x ) = 1 f ′ ⎛ ⎝ g ( x )
⎞ ⎠ , begin by finding f ′( x ).
Thus,
n ) n −1 = nx ( n −1)/ n .
⎞ ⎠ = n ( x 1/
f ′( x ) = nx n −1 and f ′ ⎛
⎝ g ( x )
Finally,
(1− n )/ n = 1
(1/ n )−1 .
g ′( x ) = 1
= 1 n x
n x
nx ( n −1)/ n
m and apply the chain rule. Thus,
To differentiate x m / n we must rewrite it as ⎛ ⎝ x 1/ n ⎞ ⎠
⎛ ⎝ ⎛
m ⎞
m −1
⎛ ⎝ x m / n
⎞ ⎠ = d
⎞ ⎠
⎛ ⎝ x 1/ n
⎞ ⎠
d dx
(1/ n )−1 = m
( m / n )−1 .
⎝ x 1/ n
⎠ = m
· 1 n x
n x
dx
□
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