300
Chapter 3 | Derivatives
Theorem 3.11: Inverse Function Theorem Let f ( x ) be a function that is both invertible and differentiable. Let y = f −1 ( x ) be the inverse of f ( x ). For all x satisfying f ′ ⎛ ⎝ f −1 ( x ) ⎞ ⎠ ≠0, dy dx = d dx ⎛ ⎝ f −1 ( x ) ⎞ ⎠ = ⎛ ⎝ f −1 ⎞ ⎠ ′( x ) = 1 f ′ ⎛ ⎝ f −1 ( x ) ⎞ ⎠ .
Alternatively, if y = g ( x ) is the inverse of f ( x ), then
g '( x ) = 1 f ′ ⎛
.
⎞ ⎠
⎝ g ( x )
Example 3.60 Applying the Inverse Function Theorem Use the inverse function theorem to find the derivative of g ( x ) = x +2
x . Compare the resulting derivative to that
obtained by differentiating the function directly.
Solution The inverse of g ( x ) = x +2 x
is f ( x ) = 2
Since g ′( x ) = 1 f ′ ⎛
⎞ ⎠ , begin by finding f ′( x ). Thus,
x −1 .
⎝ g ( x )
x 2
and f ′ ⎛
⎞ ⎠ = −2 ⎛
f ′( x ) = −2
⎝ g ( x )
= −2 ⎛ ⎝ x +2
2 = −
2 .
( x −1) 2
⎠ 2
⎝ g ( x )−1 ⎞
⎞ ⎠
x −1
Finally,
g ′( x ) = 1 f ′ ⎛
= − 2 x 2
.
⎞ ⎠
⎝ g ( x )
We can verify that this is the correct derivative by applying the quotient rule to g ( x ) to obtain g ′( x ) = − 2 x 2 .
3.42
Use the inverse function theorem to find the derivative of g ( x ) = 1 x +2 .
Compare the result obtained
by differentiating g ( x ) directly.
Example 3.61 Applying the Inverse Function Theorem
Use the inverse function theorem to find the derivative of g ( x ) = x 3 .
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