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01_apcalc4e_45342_fm_i_xxix_3pp.indd 23
Sullivan 05 apcalc4e 45342 ch03 234 283 3pp July 13, 2023 9:41
260 Chapter 3 • The Derivative of Composite, Implicit, and Inverse Functions
EXAMPLE 2 Finding an Equation of a Tangent Line
The function f (x) = x + 2x − 4 is one-to-one and has an inverse function g. Find an
3
equation of the tangent line to the graph of g at the point (8, 2) on g.
Solution
RECALL If the functions f and g are inverses The slope of the tangent line to the graph of g at the point (8, 2) is g (8). Since f and g
�
and if f (a) = b, then g(b) exists, and are inverse functions, g(8) = 2 and f (2) = 8. Now use (1) to find g (8).
�
g(b) = a.
1 1
�
g (8) = g � (y 0 ) = ; x 0 = 2; y 0 = 8
f � (2) f � (x 0 )
1
= f � (x) = 3x 2 + 2; f � (2) = 3 · 2 2 + 2 = 14
14
Now use the point-slope form of an equation of a line to find an equation of the
tangent line to g at (8, 2). 1
y − 2 = 14 (x − 8)
14y − 28 = x − 8
14y − x = 20
The line 14y − x = 20 is tangent to the graph of g at the point (8, 2).
NOW WORK Problem 51 and AP Practice Problems 7 and 11.
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2 Find the Derivative of the Inverse Trigonometric Functions
NEED TO REVIEW? Inverse trigonometric
functions are discussed in Section P.7, Table 1 lists the inverse trigonometric functions and their domains.
pp. 63--67. TABLE 1 The Domain of the Inverse Trigonometric Functions
AP EXAM TIP f Restricted Domain f −1 Domain
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The exam often uses arcsin x instead of f (x) = sin x − , f −1 (x) = sin −1 x [−1, 1]
π π
2 2
sin −1 x, arccos x instead of cos −1 x, and so on. f (x) = cos x [0,π] f −1 (x) = cos −1 x [−1, 1]
Be familiar with both notations.
π π
f (x) = tan x − , f −1 (x) = tan −1 x (−∞, ∞)
2 2
π π
f (x) = csc x −π, − ∪ 0, f −1 (x) = csc −1 x |x|≥ 1
2 2
CAUTION Keep in mind that f −1 (x) π 3π −1 −1
represents the inverse function of f f (x) = sec x 0, 2 ∪ π, 2 f (x) = sec x |x|≥ 1
and not the reciprocal function.
1 f (x) = cot x (0, π) f −1 (x) = cot −1 x (−∞, ∞)
That is, f −1 (x) �= .
f (x) π π
Sullivan 05 apcalc4e 45342 ch03 234 283 3pp July 13, 2023 9:41 To find the derivative of y = sin −1 x, −1 ≤ x ≤ 1, − 2 ≤ y ≤ 2 , we write sin y = x
and differentiate implicitly with respect to x.
d
d
Chapter 3 • The Derivative
270 NOTE Alternatively, we can use the of Composite, Implicit, and Inverse Functions dx x
Derivative of an Inverse Function.
sin y =
dx
If y = sin −1 x, then x = sin y, and
2 Use Logarithmic Differentiation d dy
dy
dy 1 1 cos y · = 1 dx sin y = cos y dx using the Chain Rule.
dx are very useful for finding derivatives of functions
= = Logarithms and their properties
dx dx cos y that involve products, quotients, or powers. This method, called logarithmic
dy dy 1
differentiation, uses the facts that =the logarithm of a product is a sum, the logarithm
dx
cos y
provided cos y �= 0. of a quotient is a difference, and the logarithm of a power is a product.
provided
y �
π π π π
, we exclude these values.
�
�
=
or y = =
provided cos y �= 0. Since cos y = 0 if y =− 2 2 or y 2 2 , we exclude these values.
provided cos y �= 0. Since cos y = 0 if y =−
EXAMPLE 4 Finding Derivatives Using Logarithmic Differentiation
CALC CLIP
x 2
Find y � if y = .
(3x − 2) 3
Solution
It is easier to find y � if we take the natural logarithm of each side before differentiating.
That is, we write
World History Archive/Alamy ln y = ln (3x − 2) 3
x 2
ORIGINS Logarithmic differentiation was and simplify using properties of logarithms.
2
first used in 1697 by Johann Bernoulli ln y = ln x − ln(3x − 2) 3
(1667--1748) to find the derivative of y = x x .
Johann, a member of a famous family of = 2 ln x − 3 ln(3x − 2)
mathematicians, was the younger brother of
Jakob Bernoulli (1654--1705). He was also a
contemporary of Newton, Leibniz, the To find y � , use implicit differentiation.
French mathematician Guillaume de
l’Hˆopital, and the Japanese mathematician dx d ln y = dx d [2 ln x − 3 ln(3x − 2)]
Seki Takakazu (1642–1708).
y � d d [3 ln(3x − 2)] d 1 dy y �
y = dx (2 ln x) − dx dx ln y = y dx = y
y � 2 9
y = x − 3x − 2
2 9 x 2 2 9
y = y − = −
�
x 3x − 2 (3x − 2) 3 x 3x − 2
Summarizing these steps, we have the method of Logarithmic Differentiation.
xxiii
© 2024 BFW Publishers PAGES NOT FINAL x x iii
Steps for Using Logarithmic Differentiation
Step 1 If the function y = f (x) consists of products, quotients, and powers, take
Do Not Copy or Post in Any Form.
the natural logarithm of each side. Then simplify using properties of
logarithms.
d y �
Step 2 Differentiate implicitly, and use the fact that ln y = .
dx y
Step 3 Solve for y � , and replace y with f (x). 10/11/23 2:38 PM
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NOW WORK Problem 51 and AP Practice Problem 12.