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Chapter 3

Differentiation Rules

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Problem 1

Explain why the natural logarithmic function $ y = \ln x $ is used much more frequently in calculus that the other logarithmic functions $ y = \log_b x. $

Carson M.
Numerade Educator

Problem 2

Differentiate the function.

$ f(x) = x \ln x - x $

Carson M.
Numerade Educator

Problem 3

Differentiate the function.

$ f(x) = \sin (\ln x) $

Doruk I.
Numerade Educator

Problem 4

Differentiate the function.

$ f(x) = \ln (\sin^2 x) $

fk
Fallo K.
Numerade Educator

Problem 5

Differentiate the function.

$ f(x) = \ln \frac {l}{x} $

Carson M.
Numerade Educator

Problem 6

Differentiate the function.

$ y = \frac{l}{\ln x} $

Doruk I.
Numerade Educator

Problem 7

Differentiate the function.

$ f(x) = \log_{10} (1 + \cos x) $

Stephanie C.
Numerade Educator

Problem 8

Differentiate the function.

$ f(x) = \log_10 \sqrt x $

Carson M.
Numerade Educator

Problem 9

Differentiate the function.

$ g(x) = \ln (xe^{-2x}) $

Carson M.
Numerade Educator

Problem 10

Differentiate the function.

$ g(t) = \sqrt {1 + \ln t} $

Doruk I.
Numerade Educator

Problem 11

Differentiate the function.

$ F(t) = (\ln t)^2 \sin t $

Carson M.
Numerade Educator

Problem 12

Differentiate the function.

$ h(x) = \ln (x + \sqrt {x^2 - 1}) $

DV
Danie V.
Numerade Educator

Problem 13

Differentiate the function.

$ G(y) = \ln \frac {(2y + 1)^5}{\sqrt {y^2 + 1}} $

Doruk I.
Numerade Educator

Problem 14

Differentiate the function.

$ P(v) = \frac {\ln v}{l - v} $

Doruk I.
Numerade Educator

Problem 15

Differentiate the function.

$ F(s) = \ln \ln s $

Doruk I.
Numerade Educator

Problem 16

Differentiate the function.

$ y = \ln \mid 1+ t - t^3 \mid $

Doruk I.
Numerade Educator

Problem 17

Differentiate the function.

$ T(z) = 2^z \log_2 z $

Carson M.
Numerade Educator

Problem 18

Differentiate the function.

$ y = \ln (\csc x - \cot x) $

Madi S.
Numerade Educator

Problem 19

Differentiate the function.

$ y = \ln (e^{-x} + xe^{-x}) $

Doruk I.
Numerade Educator

Problem 20

Differentiate the function.

$ H(z) = \ln \sqrt {\frac {a^2 - z^2}{a^2 + z^2}} $

Doruk I.
Numerade Educator

Problem 21

Differentiate the function.

$ y = \tan [\ln (ax + b)] $

Doruk I.
Numerade Educator

Problem 22

Differentiate the function.

$ y = \log_2 (x \log_5 x) $

Carson M.
Numerade Educator

Problem 23

Find $y^{\prime}$ and $y^{\prime \prime}$
$y=\sqrt{x} \ln x$

Carson M.
Numerade Educator

Problem 24

Find $ y' $ and $ y". $

$ y = \frac {\ln x}{l + \ln x} $

Doruk I.
Numerade Educator

Problem 25

Find $ y' $ and $ y". $

$ y = \ln \mid \sec x \mid $

Carson M.
Numerade Educator

Problem 26

Find $ y' $ and $ y". $

$ y = \ln (l + \ln x) $

Madi S.
Numerade Educator

Problem 27

Differentiate $ f $ and find the domain of $ f. $

$ f(x) = \frac {x}{1 - \ln (x - 1)} $

Doruk I.
Numerade Educator

Problem 28

Differentiate $ f $ and find the domain of $ f. $

$ f(x) = \sqrt {2 + \ln x} $

Doruk I.
Numerade Educator

Problem 29

Differentiate $ f $ and find the domain of $ f. $

$ f(x) = \ln (x^2 - 2x) $

Doruk I.
Numerade Educator

Problem 30

Differentiate $ f $ and find the domain of $ f. $

$ f(x) = \ln \ln \ln x $

Doruk I.
Numerade Educator

Problem 31

If $ f(x) = \ln (x + \ln x), $ find $ f'(1). $

Doruk I.
Numerade Educator

Problem 32

If $ f(x) = \cos (\ln x^2), $ find $ f'(1). $

Doruk I.
Numerade Educator

Problem 33

Find an equation of the tangent line to the curve at the given point.

$ y = \ln (x^2 - 3x + 1), (3, 0) $

Madi S.
Numerade Educator

Problem 34

Find an equation of the tangent line to the curve at the given point.

$ y = x^2 \ln x, (1, 0) $

Madi S.
Numerade Educator

Problem 35

If $ f(x) = \sin x + \ln x, $ find $ f'(x). $ Check that your answer is reasonable by comparing the graphs of $ f $ and $ f'. $

Doruk I.
Numerade Educator

Problem 36

Find equations of the tangent lines to the curve $ y = (\ln x)/x $ at the points (1, 0) and $ (e, 1/e). $ Illustrate by graphing the curve and its tangent lines.

Doruk I.
Numerade Educator

Problem 37

Let $ f(x) = cx + \ln(\cos x). $ For what value of $ c $ is $ f'(\pi/4) = 6? $

Doruk I.
Numerade Educator

Problem 38

Let $ f(x) = \log_b (3x^2 - 2). $ For what value of $ b $ is $ f'(1) = 3? $

Doruk I.
Numerade Educator

Problem 39

Use logarithmic differentiation to find the derivative of the function.

$ y = (x^2 + 2)^2 (x^4 + 4)^4 $

Doruk I.
Numerade Educator

Problem 40

Use logarithmic differentiation to find the derivative of the function.

$ y = \frac {e^{-x} \cos^2 x}{x^2 + x + 1} $

Doruk I.
Numerade Educator

Problem 41

Use logarithmic differentiation to find the derivative of the function.

$ y = \sqrt {\frac {x -1}{x^4 + 1}} $

Doruk I.
Numerade Educator

Problem 42

Use logarithmic differentiation to find the derivative of the function.

$ y = \sqrt x e^{x^2 -x}(x + 1)^{2/3} $

Bobby B.
University of North Texas

Problem 43

Use logarithmic differentiation to find the derivative of the function.

$ y = x^x $

Carson M.
Numerade Educator

Problem 44

Use logarithmic differentiation to find the derivative of the function.

$ y = x^{\cos x} $

Carson M.
Numerade Educator

Problem 45

Use logarithmic differentiation to find the derivative of the function.

$ y = x^{\sin x} $

Carson M.
Numerade Educator

Problem 46

Use logarithmic differentiation to find the derivative of the function.

$ y = (\sqrt x)^x $

Madi S.
Numerade Educator

Problem 47

Use logarithmic differentiation to find the derivative of the function.

$ y = (\cos x)^x $

Madi S.
Numerade Educator

Problem 48

Use logarithmic differentiation to find the derivative of the function.

$ y = (\sin x)^{\ln x} $

Madi S.
Numerade Educator

Problem 49

Use logarithmic differentiation to find the derivative of the function.

$ y = (\tan x)^{1/x} $

Madi S.
Numerade Educator

Problem 50

Use logarithmic differentiation to find the derivative of the function.

$ y = (\ln x)^{\cos x} $

Doruk I.
Numerade Educator

Problem 51

Find $ y' $ if $ y $ = In $ (x^2 + y^2). $

Bobby B.
University of North Texas

Problem 52

Find $ y' $ if $ x^y = y^x. $

Bobby B.
University of North Texas

Problem 53

Find a formula for $ f^{(n)}(x) $ if $ f(x) = \ln (x - 1). $

Doruk I.
Numerade Educator

Problem 54

Find $ \frac {d^9}{dx^9}(x^8 \ln x). $

Doruk I.
Numerade Educator

Problem 55

Use the definition of derivative to prove that

$ \displaystyle \lim_{x \to 0} \frac {\ln (1 + x)}{x} = 1 $

Doruk I.
Numerade Educator

Problem 56

Show that $ \displaystyle \lim_{n \to x} (1 + \frac {x}{n})^n = e^x $ for any $ x > 0. $

Doruk I.
Numerade Educator