Section 4.7: Derivatives of Inverse Functions
Theorem: (Derivative of an Inverse Function)
Suppose that f is a one-to-one differentiable function with inverse f −1 . Then
d −1 1
f (x) = 0 −1
.
dx
f [f (x)]
Example: Let f (x) =
√
x − 1.
(a) Find the inverse function, f −1 .
(b) Differentiate the inverse function directly.
(c) Differentiate the inverse using the theorem.
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Example: Let f (x) = x5 − x3 + 2x. Find (f −1 )0 (2). (Hint: f (1) = 2)
Example: Let f (x) = x − cos x. Find (f −1 )0 (−1).
Example: Let f (x) = 3 + x + ex . Find (f −1 )0 (4).
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Example: Use implicit differentiation to prove that
d
1
(ln x) = .
dx
x
Theorem: (Derivative of the Natural Logarithm)
d
1
(ln x) =
dx
x
By the Chain Rule, it follows that
f 0 (x)
d
ln(f (x)) =
.
dx
f (x)
Example: Differentiate each function.
(a) f (x) = ln(3x2 + 2x − 6)
(b) f (x) = ln(cos x)
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(c) f (x) =
√
ln x
(d) f (x) = ln(x + ln x)
(e) f (x) =
ln x
x2 + 1
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Example: Use the change of base formula to prove that
d
1
(loga x) =
.
dx
x ln a
Theorem: (Derivative of a General Logarithm)
d
1
(loga x) =
dx
x ln a
By the Chain Rule, it follows that
f 0 (x)
d
[loga (f (x))] =
.
dx
f (x) ln a
Example: Differentiate each function.
(a) f (x) = log3 (5 − x4 )
(b) f (x) = log(3x3 − x + 2)
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Example: Use implicit differentiation to find the derivative of y = sin−1 x.
Theorem: (Derivatives of Inverse Trigonometric Functions)
d
1
(sin−1 x) = √
dx
1 − x2
1
d
(cos−1 x) = − √
dx
1 − x2
1
d
(tan−1 x) = 2
dx
x +1
Example: Differentiate f (x) = tan−1 (3x2 ) + sin−1 (2x + 1) − cos−1 (ln x).
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Logarithmic Differentiation
r
3x + 2
Example: Find the derivative of f (x) = ln
.
3x − 2
Note: To compute derivatives of complicated functions involving products, quotients, or
powers can be simplified by taking logarithms. This method is known as logarithmic
differentiation.
Example: Differentiate each function using logarithmic differentiation.
(a) f (x) = xsin x
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(b) g(x) = (ln x)x
√
ex x4 + 2
(c) f (x) =
(x + 1)4 (x2 + 3)2
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