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Unformatted text preview: Lecture 28 1 Lecture 28: (Sec. 5.5)
Diﬀerentiation of Logarithmic Functions
How do we ﬁnd the derivative of
f (x) = lnx? We have: NOTE: d
(ln x) =
dx Lecture 28 2 ex. Find each xvalue at which the graph of f (x) =
xlnx has a horizontal tangent line. Then ﬁnd the
relative extrema of f . Lecture 28 3 The Chain Rule for Logarithmic Functions
Let f be a diﬀerentiable function of x. Then for
f (x) > 0,
d
[ln(f (x))] =
dx ex. d
[ln(2x − x2)] =
dx Lecture 28 4 Recall some basic Properties of the Natural Logarithm:
1) domain: 2) ln 1 = 3) ln e = 4) The Inverse Properties 5) ln (xy ) =
x
6) ln ( ) =
y
7) ln (xy ) = Lecture 28 ex. Find f (x) for each of the following:
√
1) f (x) = ln x 2) f (x) = ln(x2 + 1)3 3) f (x) = [ln(x2 + 1)]3 5 Lecture 28 6 ex. Find the equation of the tangent line to f (x) =
ln(ln x) at x = e. Lecture 28 7 Simplifying before Diﬀerentiating
ex. Find f (x) for f (x) = ln 4 2x − 1
(x + 1)3 Lecture 28 Logarithmic Diﬀerentiation
dy
x2 − 1
Find
for y = √
dx
x 2x + 4 8 Lecture 28 To ﬁnd
1) 2) 3) 4) 9 dy
by logarithmic diﬀerentiation:
dx Lecture 28 ex. Find the derivative of f (x) = xx. NOTE: 10 Lecture 28 ex. Find the derivative of
f (x) = bx, b > 0, b = 1. 11 Lecture 28 Additional Examples (Master it!)
ex. Find the slope of the tangent line to
ex + 1
f (x) = ln
at x = ln3.
x−1
e 12 Lecture 28 13 ex.√ Write the equation of the tangent line to y =
e 2x 3 x3 + 1
at x = 0.
2−1
x Lecture 28 14 x ex. Find the derivative of f (x) = xe . ...
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 Spring '08
 Smith
 Calculus, Derivative, Logarithmic Functions

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