§3.5 Implicit Differentiation and Logarithms 1 Implicit

§3.5 Implicit Diﬀerentiation and Logarithms
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Stewart and Clegg, Brief Applied Calculus (Int’l/E)
§3.5 Implicit Diﬀerentiation and Logarithms
開課班級: 統計系1A 微積分
授課教師: 吳漢銘 (淡江大學數學系專任副教授)
教學網站: http://www.hmwu.idv.tw
系級:
1
學號:
姓名:
Implicit Diﬀerentiation
1.1 The method of implicit diﬀerentiation
consists of diﬀerentiating both sides
of the equation with respect to x
and then solving the resulting equation for
dy/dx .
1.2 When diﬀerentiating expressions containing y
we must remember that y represents a function of x and so the Chain Rule
applies.
Ex. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example1,
p179)
(a) If x2 + y 2 = 25, ﬁnd dy/dx. (b) Find an equation of the tangent line to the
circle x2 + y 2 = 25 at the point (3, 4).
sol:
Calculus
October 28, 2014
§3.5 Implicit Diﬀerentiation and Logarithms
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Ex. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example2,
p180)
(a) Find dy/dx if x3 + y 3 = 6xy, . (b) Find an equation of the tangent line to the
folium of Descartes x3 + y 3 = 6xy at the point (3, 3).
sol:
2
2.1
Derivative of the Natural Logarithmic Function
d
(ln x) =
dx
1
x
proof:
2.2
d
ln u =
dx
1 du
u dx
or
d
ln g(x) =
dx
g ′ (x)
g(x)
Ex. 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example4,
p182)
If g(t) = 3t − 4 ln t, ﬁnd g ′ (t). What is the slope of the graph at t = 1.
sol:
Calculus
October 28, 2014
§3.5 Implicit Diﬀerentiation and Logarithms
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Ex. 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example5,
p182)
Diﬀerentiate y = ln(x3 + 1).
sol:
Ex. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example6,
Find
p182)
d
ln(x2 + 3ex ).
dx
sol:
Ex. 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example7,
Diﬀerentiate f (x) =
√
p183)
ln x.
sol:
Calculus
October 28, 2014
§3.5 Implicit Diﬀerentiation and Logarithms
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Ex. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (example8,
p183)
Find f ′ (x) if f (x) = ln |x|.
sol:
3
The Number e as a Limit
3.1 The Number e as a Limit
e = lim (1 + x)1/x
x→0
proof:
3.2 e ≈ 2.7182818
3.3 The Power Rule
If n is any real number and f (x) = xn , then f ′ (x) = nxn−1 .
proof:
Calculus
October 28, 2014
§3.5 Implicit Diﬀerentiation and Logarithms
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- Exercises §3.5
3-10 Find dy/dx by implicit diﬀerentiation.
5. x3 + x2 y + 4y 2 = 6
8.
√
x + y = 1 + x2 y 2
9. ex
2y
=x+y
10. y 5 + x2 y 3 = 1 + yex
2
11. If f (x) + x2 [f (x)]3 = 10 and f (1) = 2, ﬁnd f ′ (1).
18. Use implicit diﬀerentiation to ﬁnd an equation of the tangent line to the curve
y 2 (y 2 − 4) = x2 (x2 − 5) at the given point (0, −2).
21-38 Diﬀerentiate the function.
25. y = (ln x)5
26. f (x) = ln(x2 + 10)
(2t + 1)3
(3t − 1)4
√
3u + 2
34. G(u) = ln
3u − 2
31. F (t) = ln
35. y = ln(e−x + xe−x )
37. y = ln(ln x)
40. Find f ′ and f ′′ : f (x) =
ln x
x2
47. Use the change of base formula loga x =
ln x
d
1
to show that
loga x =
.
ln a
dx
x ln a
48. Find the derivative of the logarithmic function. (a) f (x) = log2 (1 − 3x) (b) f (x) =
log5 (xex )
Calculus
October 28, 2014

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