Math Problems Vault

Problem 1.13.1

(1)(2)(3)(4)

Find $f^{\prime }\left(x_{0}\right)$ if:

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(1)

$f = x^{2}, x_{0} = 0,1 ;$

(2)

$f = 2 \sin 3 x, x_{0} = \pi / 6$ ;

(3)

$f = 1+\ln 2 x, x_{0} = 1$ ;

(4)

$f = x+\operatorname {ctg} x, x_{0} = \pi / 4$ .

Problem 1.13.2

Find $f^{\prime }(x)$ and indicate the domain of existence of the derivative, if:

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(1)

$f = x^{3}+2 x$ ;

(2)

$f = 1 / x$ ;

(3)

$f = \sqrt{x}$ ;

(4)

$f = x \sqrt[3]{x}$ ;

(5)

$f = 1 /\left(1+x^{2}\right)$ ;

(6)

$f = 2^{x+1}$ ;

(7)

$f = \ln x$ ;

(8)

$f = \sin 2 x$ ;

(9)

$f = \operatorname {ctg} x+2$ ;

(10)

$f = \arcsin x$ ;

(11)

$f = \arccos 3 x$ ;

(12)

$f = 7 \operatorname {arctg}(x+1)$ .

Problem 1.13.3

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = x^{3}+x^{2}+x+1$ .

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Problem 1.13.4

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = a x^{3}+b x^{2}+c x+d$ .

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Problem 1.13.5

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = 7 x^{13}+13 x^{-7}$ .

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Problem 1.13.6

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{\ln 3}{x}+e^{2}$ .

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Problem 1.13.7

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{a}{x^{2}}+\frac{b}{x^{3}}+\frac{c}{x^{4}}$ .

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Problem 1.13.8

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \sqrt{x}+\sqrt[3]{x}+\sqrt[4]{x}$ .

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Problem 1.13.9

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{3}{5} x^{5 / 3}+x^{-2}+\frac{2}{x}$ .

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Problem 1.13.10

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = x^{3} \sqrt[3]{x^{2}}+x^{7} \sqrt[3]{x}$ .

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Problem 1.13.11

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = x^{\sqrt{5}}-x^{-\sqrt{5}}$ .

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Problem 1.13.12

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{a x+b}{c x+d}, c \neq 0$ .

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Problem 1.13.13

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{x^{2}-5 x+6}{x^{2}+x+7}$ .

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Problem 1.13.14

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{\sqrt{x}}{2+\sqrt[3]{x^{2}}}$ .

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Problem 1.13.15

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = 5 x \cos x$ .

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Problem 1.13.16

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = (x+1) \operatorname {tg} x$ .

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Problem 1.13.17

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = x^{2} \operatorname {ctg} x+2$ .

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Problem 1.13.18

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \begin{cases} \sin x / x, & x \neq 0, \\ 1, & x = 0.\end{cases} \quad$

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Problem 1.13.19

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{\sqrt{x}}{\operatorname {tg} x}$ ,

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Problem 1.13.20

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{\sin x+\cos x}{\cos x-\sin x}$ .

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Problem 1.13.21

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \operatorname {arctg} x+x+\operatorname {arcctg} x$ .

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Problem 1.13.22

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = x \arcsin x$ .

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Problem 1.13.23

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \operatorname {arctg}^{2} x$ .

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Problem 1.13.24

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{\arccos x}{\arcsin x}$ .

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Problem 1.13.25

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \ln x^{3}-\frac{9}{x}-\frac{27}{2 x^{2}}$ .

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Problem 1.13.26

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = (\sqrt{2})^{x}+(\sqrt{5})^{-x}$ .

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Problem 1.13.27

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \left(x^{2}-7 x+8\right) e^{x}$ .

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Problem 1.13.28

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = 2^{x} \ln \left|x\right|$ .

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Problem 1.13.29

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = e^{x} \log_{2} x$ .

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Problem 1.13.30

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \log_{2} x \cdot \ln x \cdot \log_{3} x$ .

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Problem 1.13.31

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \log_{x} 2$

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Problem 1.13.32

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \log_{x} 2^{x}$ .

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Problem 1.13.33

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{\arcsin x}{e^{x}}$ .

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Problem 1.13.34

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = e^{a x}(a \sin b x-b \cos b x)$ .

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Problem 1.13.35

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \operatorname {sh} x \cdot \operatorname {ch} x$ .

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Problem 1.13.36

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \operatorname {sh}^{2} x-\operatorname {ch}^{2} x$ .

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Problem 1.13.37

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{\operatorname {th} x}{\operatorname {ch}^{2} x}$ .

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Problem 1.13.38

Compute the derivative of the function $y = f(x)$ . Indicate the domain of existence of the derivative. $y = \frac{\ln x}{\operatorname {cth} x}$ .

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Problem 1.13.39

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = (x+1)(x+2)(x+3)(x+4), x_{0} = -3$ .

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Problem 1.13.40

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = (x-a)(x-b)(x-c), x_{0} = a$ .

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Problem 1.13.41

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = \frac{x-a}{x-b}, a \neq b, x_{0} = a$ .

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Problem 1.13.42

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = \left(1+a x^{b}\right)\left(1+b x^{a}\right), x_{0} = 1$ .

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Problem 1.13.43

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = x(x-1)(x-2) \ldots (x-1984)(x-1985), x_{0} = 0, x_{0} = 1985$ .

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Problem 1.13.44

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = \left(2-x^{2}\right) \cos x+2 x \sin x, x_{0} = 0$ .

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Problem 1.13.45

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = \frac{\sin x-x \cos x}{\cos x+x \sin x}, x_{0} = \frac{\pi }{2}$ .

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Problem 1.13.46

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = (a x+b) \cos x+(c x+d) \sin x, x_{0} = 0$ .

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Problem 1.13.47

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = \operatorname {arctg} x \cdot \arccos x, x_{0} = 0$

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Problem 1.13.48

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = \log_{2} x \ln 2 x, x_{0} = 1$ .

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Problem 1.13.49

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = \frac{x^{2}}{\ln x}, x_{0} = e$ .

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Problem 1.13.50

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = x^{5} e^{-x}, x_{0} = 5$ .

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Problem 1.13.51

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$ . $y = x \operatorname {sh} 2 x, x_{0} = 1$ .

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Problem 1.13.52

Find the derivative of the function. $y = (3 x-7)^{10}$ .

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Problem 1.13.53

Find the derivative of the function. $y = \frac{1}{99}(1-x)^{-99}-\frac{1}{49}(1-x)^{-98}+\frac{1}{97}(1-x)^{-97}$ .

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Problem 1.13.54

Find the derivative of the function. $y = (a+b x)^{\alpha }$ .

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Problem 1.13.55

Find the derivative of the function. $y = (0,4 \cos (8 x+5)-0,6 \sin 0,8 x)^{2}$ .

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Problem 1.13.56

Find the derivative of the function. $y = \frac{\cos 3}{2} x-\frac{1}{4} \sin (2 x+3)$ .

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Problem 1.13.57

Find the derivative of the function. $y = (a \cos x+b \sin x)^{\alpha }$ .

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Problem 1.13.58

Find the derivative of the function. $y = A e^{-k^{2} x} \sin (\omega x+\alpha )$ .

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Problem 1.13.59

Find the derivative of the function. $y = \left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^{12}$ .

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Problem 1.13.60

Find the derivative of the function. $y = \sqrt{2 x^{2}+\sqrt{x^{2}+1}}$

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Problem 1.13.61

Find the derivative of the function. $y = \sqrt[13]{9+7 \sqrt[5]{2 x}}$ .

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Problem 1.13.62

Find the derivative of the function. $y = \sqrt[3]{\frac{1-x^{3}}{1+x^{3}}}$ .

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Problem 1.13.63

Find the derivative of the function. $y = \sqrt[n]{\frac{a x+b}{c x+d}}$ .

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Problem 1.13.64

Find the derivative of the function. $y = \frac{1}{\sqrt{1+x^{4}}\left(x^{2}+\sqrt{1+x^{4}}\right)}$ .

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Problem 1.13.65

Find the derivative of the function. $y = \frac{x^{2}+4}{x \sqrt{4+\left(\left(x^{2}-4\right) /(2 x)\right)^{2}}}.$

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Problem 1.13.66

Find the derivative of the function. $y = \frac{\sqrt{x^{3}}+\sqrt{x a^{2}}-\sqrt{x^{2} a}-\sqrt{a^{3}}}{\sqrt[4]{a^{5}}+\sqrt[4]{a x^{4}}-\sqrt[4]{a^{4} x}-\sqrt[4]{x^{5}}}$ .

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Problem 1.13.67

Find the derivative of the function. $y = \cos (1 / x)$ .

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Problem 1.13.68

Find the derivative of the function. $y = \operatorname {ctg} x^{2}-(1 / 3) \operatorname {tg}^{3} 2 x$ .

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Problem 1.13.69

Find the derivative of the function. $y = \frac{1-\cos (8 x-3 \pi )}{\operatorname {tg} 2 x-\operatorname {ctg} 2 x}$ .

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Problem 1.13.70

Find the derivative of the function. $y = e^{-x^{2} / 2}$ .

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Problem 1.13.71

Find the derivative of the function. $y = 2^{\sin 2 x}$ .

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Problem 1.13.72

Find the derivative of the function. $y = \frac{1}{2} \operatorname {arctg} \frac{x}{2}-\frac{1}{3} \operatorname {arctg} \frac{x}{3}$ .

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Problem 1.13.73

Find the derivative of the function. $y = \frac{\sin^{2} x}{1+\operatorname {ctg} x}+\frac{\cos^{2} x}{1+\operatorname {tg} x}$ .

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Problem 1.13.74

Find the derivative of the function. $y = \sqrt{1+\operatorname {tg}\left(x^{2}+x^{-2}\right)}$ .

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Problem 1.13.75

Find the derivative of the function. $y = \frac{1+x^{2} \operatorname {arctg} x^{2}}{\sqrt{1+x^{4}}}$ .

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Problem 1.13.76

Find the derivative of the function. $y = \ln \ln (x/2)$

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Problem 1.13.77

Find the derivative of the function. $y = \log_2^3(2x + 3)^2$

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Problem 1.13.78

Find the derivative of the function. $y = \ln \left|\sin x\right|$

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Problem 1.13.79

Find the derivative of the function. $y = \sin \ln \left|x\right|$ .

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Problem 1.13.80

Find the derivative of the function. $y = \cos \frac{1}{\log_{2} x}$ .

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Problem 1.13.81

Find the derivative of the function. $y = 3^{\operatorname {arctg}(2 x+\pi )}$ .

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Problem 1.13.82

Find the derivative of the function. $y = \operatorname {arcctg} 2^{x}$ .

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Problem 1.13.83

Find the derivative of the function. $y = 10^{x / \log_{3} x}$ .

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Problem 1.13.84

Find the derivative of the function. $y = \frac{2}{\sqrt{31}} \operatorname {arctg} \frac{4 x-5}{\sqrt{31}}$ .

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Problem 1.13.85

Find the derivative of the function. $y = \frac{1}{\sqrt{2}} \operatorname {arctg} \frac{x^{2}-1}{\sqrt{2} x}$ .

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Problem 1.13.86

Find the derivative of the function. $y = \sqrt{\operatorname {ch} x}$ .

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Problem 1.13.87

Find the derivative of the function. $y = \operatorname {arctg} \operatorname {th} x$ .

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Problem 1.13.88

Find the derivative of the function. $y = \frac{1}{2} \operatorname {th} x+\frac{\sqrt{2}}{8} \ln \frac{1+\sqrt{2} \operatorname {th} x}{1-\sqrt{2} \operatorname {th} x}$ .

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Problem 1.13.89

Find the derivative of the function. $y = \sin \cos^{2} x \cdot \cos \sin^{2} x$ .

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Problem 1.13.90

Find the derivative of the function. $y = \operatorname {tg}^{2} x /\left(\operatorname {tg} x^{2}\right)$ .

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Problem 1.13.91

Find the derivative of the function. $y = \ln \operatorname {tg}(x / 2)-\cos x \cdot \ln \operatorname {tg} x$ .

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Problem 1.13.92

Find the derivative of the function. $y = \frac{x}{2}(\sin \ln x+\cos \ln x)$ .

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Problem 1.13.93

Find the derivative of the function. $y = \cos^{n} x \cdot \cos n x$ .

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Problem 1.13.94

Find the derivative of the function. $y = \frac{1}{2} \ln \left(1+x^{2}\right)-\frac{1}{4} \ln \left(1+x^{4}\right)-\frac{1}{2\left(1+x^{2}\right)}$ .

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Problem 1.13.95

Find the derivative of the function. $y = \ln \left(x^{2}+\sqrt{x^{4}+1}\right)$ .

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Problem 1.13.96

Find the derivative of the function. $y = 2 x \ln \left(2 x+\sqrt{4 x^{2}+1}\right)-\sqrt{4 x^{2}+1}$ .

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Problem 1.13.97

Find the derivative of the function. $y = \sin (\arcsin x)$ .

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Problem 1.13.98

Find the derivative of the function. $y = \cos (2 \arccos x)$ .

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Problem 1.13.99

Find the derivative of the function. $y = \cos (3 \arccos x)$ .

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Problem 1.13.100

Find the derivative of the function. $y = \frac{1}{\sqrt{2}} \arcsin \left(\sqrt{\frac{2}{3}} \sin x\right)$ .

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Problem 1.13.101

Find the derivative of the function. $y = \arccos \frac{x^{2 n}-1}{x^{2 n}+1}$ .

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Problem 1.13.102

Find the derivative of the function. $y = \ln \left(\frac{x+3}{x+2}\right)^{2}-\frac{2 x+5}{(x+2)(x+3)}$ .

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Problem 1.13.103

Find the derivative of the function. $y = \frac{1}{2 \sqrt{6}} \ln \left(\frac{\sqrt{2}+x \sqrt{3}}{\sqrt{2}-x \sqrt{3}}\right)^{2}$ .

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Problem 1.13.104

Find the derivative of the function. $y = \frac{1}{\sqrt{2}} \ln \left(\frac{x-\sqrt{2}}{x+\sqrt{2}}\right)^{2}-\frac{4}{\sqrt{3}} \operatorname {arctg} \frac{x}{\sqrt{3}}$ .

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Problem 1.13.105

Find the derivative of the function. $y = 2^{\sin x^{2}}$ .

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Problem 1.13.106

Find the derivative of the function. $y = 3^{\cos^{2} x}$ .

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Problem 1.13.107

Find the derivative of the function. $y = e^{\sqrt{(1-x) /(1+x)}}$ .

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Problem 1.13.108

Find the derivative of the function. $y = \operatorname {arctg} \operatorname {tg}^{2} x$ .

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Problem 1.13.109

Find the derivative of the function. $y = \log_{2} \log_{3} \log_{5} x$ .

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Problem 1.13.110

Find the derivative of the function. $y = \ln \ln \ln x^{2}$ .

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Problem 1.13.111

Find the derivative of the function. $y = \sqrt{x^{2}+1}-\ln \left(1 / x+\sqrt{1+1 / x^{2}}\right)$ .

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Problem 1.13.112

Find the derivative of the function. $y = \frac{2}{7} \ln \left(\sqrt{x^{7}}+\sqrt{1+x^{7}}\right)$ .

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Problem 1.13.113

Find the derivative of the function. $y = \operatorname {tg}^{2} x+\ln \cos^{2} x$ .

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Problem 1.13.114

Find the derivative of the function. $y = \ln (\sqrt{2} \cos x+\sqrt{\cos 2 x})$ .

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Problem 1.13.115

Find the derivative of the function. $y = e^{\sqrt{\ln \left(x^{2}+x+1\right)}}$ .

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Problem 1.13.116

Find the derivative of the function. $y = \ln \sqrt{\frac{1-\sin x}{1+\sin x}}$ .

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Problem 1.13.117

Find the derivative of the function. $y = \frac{x}{\sqrt{e^{2 x}-1}}-\operatorname {arctg} \sqrt{e^{2 x}-1}$ .

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Problem 1.13.118

Find the derivative of the function. $y = \frac{\operatorname {ch} x^{2}}{\operatorname {sh}^{2} x^{2}}-\ln \operatorname {cth} \frac{x^{2}}{2}$ .

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Problem 1.13.119

Find the derivative of the function. $y = \ln \operatorname {tg}\left(\frac{\pi }{4}+\frac{\sin x}{2}\right)$ .

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Problem 1.13.120

Find the derivative of the function. $y = \arccos \left(\sin x^{4}-\cos x^{4}\right)$ .

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Problem 1.13.121

Find the derivative of the function. $y = \frac{1}{\sin^{4} x+1}+\ln \frac{\sin^{4} x}{\sin^{4} x+1}$ .

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Problem 1.13.122

Find the derivative of the function. $y = \ln \frac{\sqrt{3}-\sqrt{2} \cos x}{\sqrt{3}+\sqrt{2} \cos x}$ .

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Problem 1.13.123

Find the derivative of the function. $y = \frac{1}{\sqrt{b^{2}-a^{2}}} \ln \frac{b+a \cos x+\sqrt{b^{2}-a^{2}} \sin x}{a+b \cos x}$ .

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Problem 1.13.124

Find the derivative of the function. $y = x / 2+\sqrt{3} \operatorname {arctg}(\operatorname {tg}(x / 2)) / \sqrt{3}$ .

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Problem 1.13.125

Find the derivative of the function. $y = \operatorname {arctg} e^{x / 2}-\ln \sqrt{e^{x} /\left(e^{x}+1\right)}$ .

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Problem 1.13.126

Find the derivative of the function. $y = \arcsin \frac{\sin \alpha \cdot \sin x}{1-\cos \alpha \cdot \cos x}$ .

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Problem 1.13.127

Find the derivative of the function. $y = \frac{1}{\sin \alpha } \ln \frac{1+x}{1-x}-\operatorname {ctg} \alpha \cdot \ln \frac{1+x \cos \alpha }{1-x \cos \alpha }$ .

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Problem 1.13.128

Find the derivative of the function. $y = \ln \sqrt{x^{2}-2 x \cos \alpha +1}+\operatorname {ctg} \alpha \cdot \operatorname {arctg} \frac{x-\cos \alpha }{\sin \alpha }$ .

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Problem 1.13.129

Find the derivative of the function. $y = x^{2} \sqrt{a^{2}+x^{4}}+a^{2} \ln \left(x^{2}+\sqrt{a^{2}+x^{4}}\right)$ .

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Problem 1.13.130

Find the derivative of the function. $y = \operatorname {th} x+\frac{\sqrt{2}}{4} \ln \frac{1+\sqrt{2} \operatorname {th} x}{1-\sqrt{2} \operatorname {th} x}$ .

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Problem 1.13.131

Find the derivative of the function. $y = \ln \frac{x^{2}+a}{\sqrt{x^{4}+b^{2}}}+\frac{a}{b} \operatorname {arctg} \frac{x^{2}}{b}$ .

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Problem 1.13.132

Find the derivative of the function. $y = x \sqrt{a^{2}-x^{2}}+a^{2} \arcsin \frac{x}{\left|a\right|}$ .

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Problem 1.13.133

Find the derivative of the function. $y = x+\operatorname {ctg} x \cdot \ln (1+\sin x)-\ln \operatorname {tg}(x / 2)$ .

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Problem 1.13.134

Find the derivative of the function. $y = x-\ln \sqrt{1+e^{2 x}}+e^{-x} \operatorname {arcctg} e^{x}$ .

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Problem 1.13.135

Find the derivative of the function. $y = \ln \frac{1-\sqrt[3]{x^{2}}}{\sqrt{1+\sqrt[3]{x^{2}}+\sqrt[3]{x^{4}}}}+\sqrt{3} \operatorname {arctg} \frac{1+2 \sqrt[3]{x^{2}}}{\sqrt{3}}$ .

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Problem 1.13.136

Find the derivative of the function. $y = \ln \sqrt{\frac{\sqrt{x^{4}+1}-\sqrt{2} x}{\sqrt{x^{4}+1}+\sqrt{2} x}}-\operatorname {arctg} \frac{\sqrt{2} x}{\sqrt{x^{4}+1}}$ .

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Problem 1.13.137

Find the derivative of the function. $y = \ln \frac{2 x^{2}+4 x+4}{2 x^{2}+2 x+1}+4 \operatorname {arctg}(x+1)-\operatorname {arctg}(2 x+1)$ .

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Problem 1.13.138

Find the derivative of the function. $y = \frac{5 x+2}{x^{2}+x+1}+\ln \sqrt[3]{\frac{(x-1)^{2}}{x^{2}+x+1}}+\frac{8}{\sqrt{3}} \operatorname {arctg} \frac{2 x+1}{\sqrt{3}}$ .

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Problem 1.13.139

Find the derivative of the function. $y = \frac{3-\sin x}{2} \sqrt{\cos^{2} x-2 \sin x}+2 \arcsin \frac{1+\sin x}{\sqrt{2}}$ .

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Problem 1.13.140

Find the derivative of the function. $y = e^{x} \arcsin \sqrt{e^{x} /\left(e^{x}+1\right)}+\operatorname {arctg} \sqrt{e^{x}}-\sqrt{e^{x}}$ .

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Problem 1.13.141

Find the derivative of the function. $y = x^{x}$ .

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Problem 1.13.142

Find the derivative of the function. $y = x^{7 / \ln x}$ .

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Problem 1.13.143

Find the derivative of the function. $y = \log_{x} 7$ .

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Problem 1.13.144

Find the derivative of the function. $y = \log_{2} x \cdot \log_{x} e+\log_{2} x \cdot \ln 2$ .

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Problem 1.13.145

Find the derivative of the function. $y = x^{2 / \ln x}-2 x^{\log_{x} e} e^{1+\ln x}+e^{1+2 / \log_{x} e}$ .

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Problem 1.13.146

Find the derivative of the function. $y = x^{x^{2}}$ .

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Problem 1.13.147

Find the derivative of the function. $y = x^{e^{x}}$ .

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Problem 1.13.148

Find the derivative of the function. $y = 2^{x^{x}}$ .

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Problem 1.13.149

Find the derivative of the function. $y = x^{x^{x}}$ .

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Problem 1.13.150

Find the derivative of the function. $y = |\sin x|^{\cos x}$ .

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Problem 1.13.151

Find the derivative of the function. $y = \left(\arcsin \sin^{2} x\right)^{\operatorname {arctg} x}$ .

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Problem 1.13.152

Find the derivative of the function. $y = (\operatorname {ch} x)^{e^{x}}$ .

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Problem 1.13.153

Compute the derivative of the function at the indicated points. $y = \frac{1+x-x^{2}}{1-x+x^{2}}, x = 0, x = 1$ .

?

Problem 1.13.154

Compute the derivative of the function at the indicated points. $y = (1+x) \sqrt{2+x^{2}} \sqrt[3]{3+x^{3}}, x = 0$ .

?

Problem 1.13.155

Compute the derivative of the function at the indicated points. $y = 2^{\operatorname {tg}(1 / x)}, x = 1 / \pi$ .

?

Problem 1.13.156

Compute the derivative of the function at the indicated points. $y = 3 \cos 2 x-\sqrt{1-\sin 2 x}(\sin x+\cos x), x = \pi / 6$ .

?

Problem 1.13.157

Compute the derivative of the function at the indicated points. $y = \log_{1 / 2}(x-1 / 2)^{2}+\log_{2} \sqrt{4 x^{2}-4 x+1}, x = 0$ .

?

Problem 1.13.158

Compute the derivative of the function at the indicated points. $y = \sqrt{\ln x}\left(\ln x-\log_{e x} x\right) \sqrt{\ln x+\log_{x} e+2}, x = e$ .

?

Problem 1.13.159

Compute the derivative of the function at the indicated points. $y = \ln \left(1+\sin^{2} x\right)-2 \sin x \cdot \operatorname {arctg} \sin x, x = \pi / 2$ .

?

Problem 1.13.160

Compute the derivative of the function at the indicated points. $y = \arcsin \left(2 x /\left(1+x^{2}\right)\right), x = 0, x = 2$ .

?

Problem 1.13.161

Compute the derivative of the function at the indicated points. $y = \arcsin \left(\left(1-x^{2}\right) /\left(1+x^{2}\right)\right), x = -1, x = 1, x = 0$ .

?

Problem 1.13.162

Compute the derivative of the function at the indicated points. $y = \ln \frac{x^{4}-x^{2}+1}{x^{4}+2 x^{2}+1}+2 \sqrt{3} \operatorname {arctg} \frac{\sqrt{3}}{1-2 x^{2}}, x = 1$ .

?

Problem 1.13.163

Compute the derivative of the function at the indicated points. $y = \sqrt[3]{\operatorname {arctg} \sqrt[5]{\cos \ln^{3} x}}, x = 1$ .

?

Problem 1.13.164

Compute the derivative of the function at the indicated points. $y = (\operatorname {ch} x)^{\operatorname {sh} x}, x = 0$ .

?

Problem 1.13.165

Compute the derivative of the function at the indicated points. $y = \left(\sqrt{1+3^{x}}\right)^{\ln x^{2}}, x = 1$ .

?

Problem 1.13.166

Compute the derivative of the function at the indicated points. $y = ((\sin x) / x)^{x}, x = \pi / 2$ .

?

Problem 1.13.167

Solve the equation $y^{\prime }(x) = 0$ if:

?

(1)

$y = x^{3}-6 x^{2}+9 x+12$ ;

(2)

$y = \frac{x^{2}+x-6}{x^{2}-10 x+25}$ ;

(3)

$y = \frac{1}{1+\sin^{2} x}$ ;

(4)

$y = x(x-1)^{2}(x-2)^{3}$ ;

(5)

$y = e^{-\left|x-1\right|} /(1+x)$ ;

(6)

$y = \max \left(7 x-6 x^{2},|x|^{3}\right)$ .

Problem 1.13.168

(1)(2)(3)(4)

Let $f(x), x \in \mathbb {R}$ , be an everywhere differentiable function. Find $y^{\prime }(x)$ if:

?

(1)

$y = \sqrt{f(x)}, f(x) > 0$

(2)

$y = \ln \left|f(x)\right|, f(x) \neq 0$

(3)

$y = f\left(x^{3}\right)$

(4)

$y = f(\arcsin f(x)),\left|f(x)\right| < 1$ .

Problem 1.13.169

(1)(2)(3)(4)

Let $f(x), x \in \mathbb {R}$ , and $g(x), x \in \mathbb {R}$ , be everywhere differentiable functions. Find $y^{\prime }(x)$ if:

?

(1)

$y = \sqrt[n]{f^{2}(x)+g^{2}(x)}, f^{2}(x)+g^{2}(x) > 0$

(2)

$y = \ln \left|f(x) / g(x)\right|, \quad f(x) g(x) \neq 0$ ;

(3)

$y = f\left(\sin^{2} x\right)+g\left(\cos^{2} x\right)$ ;

(4)

$y = (f(x))^{g(x)}, f(x) > 0$ .

Problem 1.13.170

Let the functions $f_{i j}(1 \leqslant i, j \leqslant n)$ be differentiable at some point. Prove that at this point the equality holds

\left|\begin{array}{ccccc} f_{11} & f_{12} & \ldots & f_{1 n} \\ \ldots & \ldots & \ldots & \ldots \\ f_{i 1} & f_{i 2} & \ldots & f_{i n} \\ \ldots & \ldots & \ldots & \ldots \\ f_{n 1} & f_{n 2} & \ldots & f_{n n} \end{array}\right| = \sum _{i = 1}^{n}\left|\begin{array}{ccccc} f_{11} & f_{12} & \ldots & f_{1 n} \\ \ldots & \ldots & \ldots & \ldots \\ f_{i 1}^{\prime } & f_{i 2}^{\prime } & \ldots & f_{i n}^{\prime } \\ \ldots & \ldots & \ldots & \ldots \\ f_{n 1} & f_{n 2} & \ldots & f_{n n} \end{array}\right|.

?

Problem 1.13.171

(1)(2)(3)

Using the formula given in the preceding problem, find $\Delta^{\prime }(x)$ if:

?

(1)

$\Delta (x) = \left|\begin{array}{ccc}1 & 2 & 3 \\ x & 1 & 2 \\ x & x & 1\end{array}\right|$ ;

(2)

$\Delta (x) = \left|\begin{array}{ccc}x-1 & 2 & 3 \\ 4 & x-5 & 6 \\ 7 & 8 & x-9\end{array}\right|$ ;

(3)

$\Delta (x) = \left|\begin{array}{ccc}x & 1 & 0 \\ x^{2} & 2 x & 2 \\ x^{3} & 3 x^{2} & 6 x\end{array}\right|$ .

Problem 1.13.172

(1)(2)

Prove or disprove the following statements:

?

(1)

if a function has a derivative at some point, then it is continuous at that point;

(2)

if a function is continuous at some point, then it has a derivative at that point.

Problem 1.13.173

(1)(2)(3)

For what values of $\alpha$ does the function

y = \begin{cases} |x|^{\alpha } \sin (1 / x), & \text{ if } & x \neq 0, \\ 0, & \text{ if } \quad x = 0, \end{cases}

at the point $x = 0$ :

?

(1)

is continuous;

(2)

has a derivative;

(3)

have a continuous derivative?

Problem 1.13.174

(1)(2)(3)

For what values of $\alpha$ and $\beta (\beta > 0)$ does the function

y = \begin{cases} |x|^{\alpha } \sin \left(1 /|x|^{\beta }\right), & \text{ if } \quad x \neq 0 \\ 0, & \text{ if } x = 0 \end{cases}

at the point $x = 0$ :

?

(1)

is continuous;

(2)

has a derivative;

(3)

have a continuous derivative?

Problem 1.13.175

Construct an example of a function having a derivative everywhere except at the points $x_{1}, x_{2}, \ldots , x_{n}$ .

?

Problem 1.13.176

(1)(2)(3)(4)

Determine the values of $\alpha$ and $\beta$ for which: a) the following functions are everywhere continuous; b) everywhere differentiable:

?

(1)

$y = \begin{cases} \alpha x+\beta , & \text{ if } x \leqslant 1, \\ x^{2}, & \text{ if } x > 1 ;\end{cases}$

(2)

$y = \begin{cases} \alpha +\beta x^{2}, & \text{ if }\left|x\right| \leqslant 1, \\ 1 /\left|x\right|, & \text{ if }\left|x\right| > 1 ; \end{cases}$

(3)

$y = \begin{cases} \alpha x^{3}+\beta x, & \left|x\right| \leqslant 2, \\ (1 / \pi ) \arcsin (1 / x), & \left|x\right| > 2 \end{cases}$

(4)

$y = \begin{cases} 2 x-2, & \text{ if } & x \leqslant 1, \\ \alpha (x-1)(x-2)(x-\beta ), & \text{ if } & 1 < x < 2, \\ x / 2-1, & \text{ if } & x \geqslant 2. \end{cases}$

Problem 1.13.177

(1)(2)

Determine the values of $\alpha$ and $\beta$ for which the functions are everywhere differentiable:

?

(1)

$y = \begin{cases} (x+\alpha ) e^{-\beta x}, & \text{ if } x < 0, \\ \alpha x^{2}+\beta x+1, & \text{ if } x \geqslant 0 ; \end{cases}$

(2)

$y = \begin{cases} \alpha x+\beta , & \text{ if } x < 0, \\ \alpha \cos x+\beta \sin x, & \text{ if } x \geqslant 0. \end{cases}$

Problem 1.13.178

(1)(2)

Determine the values of $\alpha$ and $\beta$ for which the function

y = \begin{cases} \operatorname {arctg} \alpha x, & \text{ if } & \left|x\right| \leqslant 1, \\ \beta \operatorname {sign} x+\frac{x-1}{2}, & \text{ if } & \left|x\right| > 1, \end{cases}

has a derivative:

?

(1)

at the point $x = 1$ ;

(2)

at the point $x = -1$ .

Problem 1.13.179

Investigate the following functions for differentiability:

?

(1)

$y = \left|x^{3}(x+1)^{2}(x+2)\right|$ ;

(2)

$y = \left|\sin x\right|$

(3)

$y = x\left|x\right|$ ;

(4)

$y = \left|\pi -x\right| \sin x$

(5)

$y = \arccos (\cos x)$

(6)

$y = \begin{cases} x^{3}, & \text{ if } x \leqslant 0 \\ e^{-1 / x}, & \text{ if } x > 0 \end{cases}$

(7)

$y = \begin{cases} x^{2}\left|\cos (\pi / x)\right|, & \text{ if } x \neq 0, \\ 0, & \text{ if } x = 0 ; \end{cases}$

(8)

$y = \begin{cases} x, & \text{ if } x-\text{ rational number, } \\ 0, & \text{ if } x-\text{ irrational number. }\end{cases}$

Problem 1.13.180

(1)(2)

Compute the values of the derivative of the function at all points where the derivative exists:

?

(1)

$y = \begin{cases} x^{2}, & \text{ if } x \text{ — rational number, } \\ 0, & \text{ if } x \text{ — irrational number. } \end{cases}$

(2)

$y = \begin{cases} x^{2}, & \text{ if } x \text{ — rational number, } \\ 2\left|x\right|-1, & \text{ if } x \text{ — irrational number. } \end{cases}$

Problem 1.13.181

Is the following statement true: if a function has a derivative at a point, then it is differentiable in some neighborhood of that point?

?

Problem 1.13.182

(1)(2)(3)(4)

Prove or disprove the following statements:

?

(1)

if a function $f$ has a derivative at some point, and a function $g$ does not, then the function $f+g$ does not have a derivative at that point;

(2)

if the functions $f$ and $g$ do not have a derivative at some point, then the function $f+g$ also does not have a derivative at that point;

(3)

if a function $f$ has a derivative at some point, and a function $g$ does not, then the function $fg$ also does not have a derivative at that point;

(4)

if the functions $f$ and $g$ do not have a derivative at some point, then the function $fg$ also does not have a derivative at that point.

Problem 1.13.183

Give an example of a function $f(x)$ such that $f(x)$ and $(f(x))^{3}$ are differentiable at the point $x_{0}$ , while the function $(f(x))^{2}$ does not have a derivative at the point $x_{0}$ .

?

Problem 1.13.184

Give an example of a function having no derivative at any point $x \in \mathbb {R}$ , whose square has a derivative at every point $x \in \mathbb {R}$ .

?

Problem 1.13.185

(1)(2)(3)

Give an example of a composite function $f(g(x))$ having a derivative at the point $x_{0}$ and such that:

?

(1)

$f^{\prime }\left(g\left(x_{0}\right)\right)$ exists, $g^{\prime }\left(x_{0}\right)$ does not exist;

(2)

$f^{\prime }\left(g\left(x_{0}\right)\right)$ does not exist, $g^{\prime }\left(x_{0}\right)$ exists;

(3)

$f^{\prime }\left(g\left(x_{0}\right)\right)$ and $g^{\prime }\left(x_{0}\right)$ do not exist.

Problem 1.13.186

(1)(2)(3)

Prove or disprove the following statements:

?

(1)

if for functions $f$ and $g$ differentiable on the interval $(a ; b)$ the inequality $f < g$ holds, then $f^{\prime } \leqslant g^{\prime }$ on $(a ; b)$ ;

(2)

if on the interval $(a ; b)$ the inequality $f^{\prime } < g^{\prime }$ holds, then $f < g$ on $(a ; b)$ ;

(3)

if $f(a) = g(a)$ and $f^{\prime }(x) < g^{\prime }(x)$ on the interval $(a ; b)$ , then $f(x) < g(x)$ on $(a ; b)$ .

Problem 1.13.187

(1)(2)(3)(4)

Prove or disprove the following statements:

?

(1)

for a differentiable function $y(x), x \in (a ; b)$ , to have a derivative that is monotonic on the interval $(a ; b)$ , it is necessary that $y(x)$ be monotonic on $(a ; b)$ ;

(2)

for a differentiable function $y(x), x \in (a ; b)$ , to have a derivative that is monotonic on the interval $(a ; b)$ , it is sufficient that $y(x)$ be monotonic on the interval $(a ; b)$ ;

(3)

for a differentiable function to have a periodic derivative, it is necessary that the function be periodic;

(4)

for a differentiable function to have a periodic derivative, it is sufficient that the function be periodic.

Problem 1.13.188

(1)(2)(3)(4)

Prove or disprove the following statements:

?

(1)

for the derivative of a differentiable function to be an even function, it is sufficient that the function be odd;

(2)

for the derivative of a differentiable function to be an even function, it is necessary that the function be odd;

(3)

for the derivative of a differentiable function to be an odd function, it is necessary that the function be even;

(4)

for the derivative of a differentiable function to be an odd function, it is sufficient that the function be even.

Problem 1.13.189

Is the following statement true: if a function $f$ has a derivative at the point $x_{0}$ , then the sequence $\left\{ n\left(f\left(x_{0}+1 / n\right)-f\left(x_{0}\right)\right)\right\}$ converges? Is the converse statement true?

?

Problem 1.13.190

(1)(2)(3)(4)

Are the following statements true:

?

(1)

if the function $y(x)$ is differentiable on the interval $(a ; b)$ and $\lim_{x \rightarrow a+0} y^{\prime }(x) = \infty$ , then $\lim_{x \rightarrow a+0} y(x) = \infty$ ;

(2)

if the function $y(x)$ is differentiable on the interval $(a ; b)$ and $\lim_{x \rightarrow a+0} y(x) = \infty$ , then $\lim_{x \rightarrow a+0} y^{\prime }(x) = \infty$ ;

(3)

if the function $y(x)$ is differentiable on the interval $(a ;+\infty )$ and $\lim_{x \rightarrow +\infty } y(x)$ exists, then $\lim_{x \rightarrow +\infty } y^{\prime }(x)$ also exists;

(4)

if the function $y(x)$ is differentiable on the interval $(a ;+\infty )$ and $\lim_{x \rightarrow +\infty } y^{\prime }(x)$ exists, does a finite or infinite $\lim_{x \rightarrow +\infty } y(x)$ exist?

Problem 1.13.191

Find the right- and left-hand derivatives at the indicated points for the function $f(x)$ if:

?

(1)

$f(x) = \left|x\right|, x = 0$ ;

(2)

$f(x) = \left|x^{2}-5 x+6\right|, x = 2, x = 3$ ;

(3)

$f(x) = \left|2^{x}-2\right|, x = 1$ ;

(4)

$f(x) = \sqrt[3]{\sin \pi x}, x = k, k \in \mathbb {Z}$ ;

(5)

$f(x) = \sqrt{\sin x^{2}}, x = 0, x = \sqrt{\pi }$

(6)

$f(x) = \sin x\left|\cos x\right|+\cos x\left|\sin x\right|, x = \pi k / 2, k \in \mathbb {Z}$ ;

(7)

$f(x) = x\left|\cos (\pi / x)\right|, x = 2 /(2 k+1), k \in \mathbb {Z}$ ;

(8)

$f(x) = \arccos (1 / x), x = -1, x = 1$

(9)

$f(x) = \arcsin \sin x, x = (2 k+1) \pi / 2, k \in \mathbb {Z}$ .

Problem 1.13.192

(1)(2)(3)

Find $f_{-}^{\prime }(0)$ and $f_{+}^{\prime }(0)$ for the function $f(x)$ if:

?

(1)

$f(x) = \begin{cases} x, & \text{ if } x \leqslant 0, \\ \sqrt[3]{x^{4}} \ln x, & \text{ if } x > 0 ; \end{cases}$

(2)

$f(x) = \begin{cases} 2 x, & \text{ if } \quad x < 0 \\ \ln \left(1+\sqrt[5]{x^{7}}\right), & \text{ if } \quad x \geqslant 0 \end{cases}$

(3)

$f(x) = \begin{cases} 1+e^{1 / x}, & \text{ if } x < 0, \\ \sqrt{1+\sqrt[3]{x^{4}}}, & \text{ if } x \geqslant 0. \end{cases}$

Problem 1.13.193

Find the right- and left-hand derivatives at the points of discontinuity of the function $f(x)$ if:

?

(1)

$f(x) = \begin{cases} x\left(1-x^{2}\right) /\left|x\right|, & \text{ if } x \neq 0 \\ 1, & \text{ if } x = 0 \end{cases}$

(2)

$f(x) = \left(1-x^{2}\right) \operatorname {sign} x$

(3)

$f(x) = \begin{cases} \frac{1}{1+e^{1 / x}}, & \text{ if } & x \neq 0 \\ 0, & \text{ if } & x = 0 \end{cases}$

(4)

$f(x) = \begin{cases} \operatorname {arctg}((1+x) /(1-x)), & \text{ if } x \neq 0, \\ \pi / 2, & \text{ if } x = 1 ; \end{cases}$

(5)

$f(x) = \begin{cases} \operatorname {arctg}(1 / x), & \text{ if } x \neq 0, \\ -\pi / 2, & \text{ if } x = 0. \end{cases}$

Problem 1.13.194

Find $f_{+}^{\prime }\left(x_{0}\right)$ and $f_{-}^{\prime }\left(x_{0}\right)$ for the function $f(x) = \left|x-x_{0}\right| \varphi (x)$ , where $\varphi (x)$ is a given function, continuous at the point $x_{0}$ .

?

Problem 1.13.195

Give an example of a function having an infinite negative derivative at a point of discontinuity.

?

Problem 1.13.196

Give an example of a function that is continuous at some point and has neither a left- nor a right-hand derivative at that point.

?

Problem 1.13.197

Find the derivative of the inverse function at the indicated points:

?

(1)

$y = x+x^{5} / 5, y = 0, y = 6 / 5$ ;

(2)

$y = 2 x-\cos x / 2, y = -1 / 2$ ;

(3)

$y = 0,1 x+e^{0,1 x}, y = 1$ ;

(4)

$y = 2 x^{2}-x^{4}, x > 1, y = 0$ ;

(5)

$y = 2 x^{2}-x^{4}, 0 < x < 1, y = 3 / 4$ .

Problem 1.13.198

(1)(2)(3)(4)

Find the derivative of the inverse function. Indicate the domain of existence of the derivative:

?

(1)

$y = x+\ln x, x > 0$

(2)

$y = x+e^{x}$ ;

(3)

$y = x^{2} /\left(1+x^{2}\right), x < 0$ ;

(4)

$y = \operatorname {ch} x, x > 0$ .

Problem 1.13.199

Find the derivative of the function inverse to the hyperbolic tangent.

?

Problem 1.13.200

Given the function $y = x+\sin x, x \in \mathbb {R}$ . At which points does the inverse function have an infinite positive derivative?

?

Problem 1.13.201

Find $y_{x}^{\prime }$ for the function $y = y(x)$ given parametrically:

?

(1)

$x = \sin^{2} t, y = \cos^{2} t, 0 < t < \pi / 2$

(2)

$x = e^{-t}, y = t^{3},-\infty < t < +\infty$ ;

(3)

$x = a \cos t, y = b \sin t, 0 < t < \pi$

(4)

$x = a \operatorname {ch} t, y = b \operatorname {sh} t,-\infty < t < 0$

(5)

$x = t^{2}+6 t+5, y = \left(t^{3}-54\right) / t, 0 < t < +\infty$ ;

(6)

$x = (t-1)^{2}(t-2), y = (t-1)^{2}(t-3), 5 / 3 < t < +\infty$ ;

(7)

$x = a(t-\sin t), y = a(1-\cos t),-\infty < t < +\infty$ ;

(8)

$x = \ln \sin (t / 2), y = \ln \sin t, 0 < t < \pi$ .

Problem 1.13.202

(1)(2)(3)

Find $x_{y}^{\prime }$ for the function $x = x(y)$ given parametrically:

?

(1)

$x = t+2 t^{2}+t^{3}, y = -2+3 t-t^{3}, 1 < t < +\infty$ ;

(2)

$x = \left(t^{3}-2 t^{2}+3 t-4\right) e^{t}, y = \left(t^{3}-2 t^{2}+4 t-4\right) e^{t}, 1 < t < +\infty$ ;

(3)

$x = \operatorname {ctg} 2 t, y = (2 \cos 2 t-1) /(2 \cos t), 0 < t < \pi / 2$ .

Problem 1.13.203

For the function $y = y(x)$ given parametrically by $x = 2 t+\left|t\right|$ , $y = 5 t^{2}+4 t\left|t\right|,-\infty < t < \infty$ , compute the derivative at the point $x = 0$ .

?

Problem 1.13.204

Find $y_{x}^{\prime }$ for the function $y = y(x)$ given parametrically:

x = \begin{cases} t, & \text{ if } t-\text{ rational number, } \\ -t, & \text{ if } t \text{ — irrational number, } \end{cases} \quad y = t^{2}.

?

Problem 1.13.205

(1)(2)(3)

Compute $y^{\prime }\left(x_{0}\right)$ for the function $y = y(x)$ given by the equation $r = r(\varphi )$ , where $r$ and $\varphi$ are the polar coordinates of the point $(x ; y)$ :

?

(1)

$r = a \varphi , 4 \pi / 3 < \varphi < 2 \pi , x_{0} = 0$

(2)

$r = e^{\varphi },-\pi / 6 < \varphi < \pi / 6, x_{0} = 1$ ;

(3)

$r = a \sqrt{\cos 2 \varphi }, 0 < \varphi < \pi / 4, x_{0} = a \sqrt{6} / 4$ .

Problem 1.13.206

For the function $y = y(x)$ given by the equation $r = a \sqrt{\cos 2 \varphi }$ , $0 \leqslant \varphi \leqslant \pi / 4$ , compute $y_{+}^{\prime }(0)$ and $y_{-}^{\prime }(a)$ .

?

Problem 1.13.207

Find $y^{\prime }$ for the differentiable function $y = y(x)$ given implicitly by the equation:

?

(1)

$y^{5}+y^{3}+y-x = 0$ ;

(2)

$y-x = \varepsilon \sin y,\left|\varepsilon \right| < 1$ ;

(3)

$y^{2} = 2 p x, y > 0$ ;

(4)

$x^{2} / a^{2}-y^{2} / b^{2} = 1, y > 0$ ;

(5)

$(2 a-x) y^{2} = x^{3}, y < 0$

(6)

$\sqrt{x}+\sqrt{y} = 2$

(7)

$x^{2 / 3}+y^{2 / 3} = a^{2 / 3}, y > 0$

(8)

$5 x^{2}+9 y^{2}-30 x+18 y+9 = 0, y < -1$ ;

(9)

$x^{2}-4 x y+4 y^{2}+4 x-3 y-7 = 0 ; x < 2 y-1$ .

Problem 1.13.208

(1)(2)(3)(4)

For the differentiable function $y = y(x)$ given implicitly, compute $y^{\prime }\left(x_{0}\right)$ :

?

(1)

$x^{2}+y^{2}-6 x+10 y-2 = 0, y > -5, x_{0} = 0$

(2)

$6 x y+8 y^{2}-12 x-26 y+11 = 0, y < 2, x_{0} = 11 / 12$ ;

(3)

$e^{y}+x y = e, y > 0, x_{0} = 0$ ;

(4)

$x y+\ln y = 1, y < e^{2}, x_{0} = 0$ .

Problem 1.13.209

(1)(2)

Find $A\left(x_{0}\right)$ and $\alpha (\Delta x)$ in formula (5) if:

?

(1)

$y = x^{3}-2 x, x_{0} = 1$ ;

(2)

$y = x^{10}, x_{0} = 0$ .

Problem 1.13.210

Find the difference between the increment and the differential of the function $y = (x-1)^{3}$ .

?

Problem 1.13.211

For what values of $x$ is the differential of the function $y = \cos x$ not equivalent, as $\Delta x \rightarrow 0$ , to its increment?

?

Problem 1.13.212

What is the order, as $\Delta x \rightarrow 0$ , of the infinitesimal $\Delta y-d y$ if $y = x^{3}-3 x$ ?

?

Problem 1.13.213

Find the differential:

?

(1)

$d\left(e^{-x}+\ln x\right)$ ;

(2)

$d(\sqrt{x}+2 \sqrt{x+\sqrt{x}})$ ;

(3)

$d\left(2 \sqrt{x^{3}}(3 \ln x-2)\right)$ ;

(4)

$d\left(\arccos e^{x}\right)$ ;

(5)

$d \ln (\sqrt{1+2 \sin x}+\sqrt{2 \sin x-1})$ ;

(6)

$d\left(5 \operatorname {sh}^{7}\left(\frac{x}{35}\right)+7 \operatorname {sh}^{5}\left(\frac{x}{35}\right)\right)$ ;

(7)

$d\left(\frac{\arcsin x}{\sqrt{1-x^{2}}}+\ln \sqrt{\frac{1-x}{1+x}}\right)$ ;

(8)

$d\left(\ln \frac{1+\sqrt{\sin x}}{1-\sqrt{\sin x}}+2 \operatorname {arctg} \sqrt{\sin x}\right)$ ;

(9)

$d\left(x^{x^{2}}\right)$ .

Problem 1.13.214

(1)(2)(3)(4)

Find the differential at the indicated points:

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(1)

$d\left(\frac{1}{x}+\ln \frac{x-1}{x}\right), x = -1$ ;

(2)

$d \operatorname {arctg} \frac{\ln x}{x}, x_{1} = \frac{1}{e}, x_{2} = e$ ;

(3)

$d\left(\frac{(2 x-1)^{3} \sqrt{2+3 x}}{(5 x+4)^{2} \sqrt[3]{1-x}}\right), x = 0$

(4)

$d\left(\frac{x^{2} 2^{x}}{x^{x}}\right), x_{1} = 1, x_{2} = 2$ .

Problem 1.13.215

At the indicated points, find the differential of the function $y = y(x)$ given by an implicit or parametric equation:

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(1)

$y^{3}-y = 6 x^{2},(1 ; 2)$ ;

(2)

$x^{4}+y^{4}-8 x^{2}-10 y^{2}+16 = 0,(1 ; 3)$ ;

(3)

$y^{5}+x^{4} = x y^{2},\left(x_{0} ; y_{0}\right)$ ;

(4)

$x+y \ln y = 0,\left(x_{0} ; y_{0}\right)$ ;

(5)

$x y-\sqrt[3]{x y^{2}+6} = 0,(2 ; 1)$ ;

(6)

$x e^{\left(x / y^{2}-1\right)}-2 y = 0,(4 ; 2)$ ;

(7)

$3^{\sin y x^{2}}-3 x(y-\pi )-1 = 0,(1 ; \pi )$ ;

(8)

$4 x y^{3}+\ln \sqrt[3]{x /(x+y)} = 0,(1 ; 0)$

(9)

$x = (t-1)^{2}(t-2), y = (t-1)^{2}(t-3),(4 ; 0)$ ;

(10)

$x = e^{t} / t, y = (t-1)^{2} e^{t},(-2 / \sqrt{e} ; 9 /(4 \sqrt{e}))$ .

Problem 1.13.216

At the point $(0 ; a)$ find the differential of the function $y = y(x)$ given in the polar coordinate system by the equation $r = a(1+\cos \varphi )$ , $0 < \varphi < \pi$ .

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Problem 1.13.217

Find the differential of the function $y$ , assuming the differentials of the functions $u$ and $v$ are known:

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(1)

$y = u^{2} v$

(2)

$y = u^{2} / v$

(3)

$y = u v /\left(u^{2}+v^{2}\right)$

(4)

$y = e^{u v}$

(5)

$y = \sqrt{u^{2}+v^{2}}$ ;

(6)

$y = \ln \operatorname {tg}(v / u)$

(7)

$y = u^{v}$ .

Problem 1.13.218

Replacing the increment of the function by its differential, find the approximate value of the function $y = y(x)$ at the indicated points:

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(1)

$y = \sqrt[3]{x}$ a) $x = 65$ b) $x = 125,1324$

(2)

$y = \sqrt[4]{x}$ a) $x = 90$ b) $x = 15,8$

(3)

$y = \sin x$ a) $x = 29^{\circ }$ b) $x = 359^{\circ }$

(4)

$y = \operatorname {tg} x, x = 44^{\circ } 50^{\prime }$

(5)

$y = \arcsin x, x = 0,51$ ;

(6)

$y = \operatorname {arctg} x, x = 1,05$ ;

(7)

$y = \ln \operatorname {tg} x, x = 47^{\circ } 15^{\prime }$ ;

(8)

$y = \sqrt{\frac{2-x}{2+x}}, x = 0,15$ .

Problem 1.13.219

(1)(2)(3)(4)

Prove that for all values of $\Delta x$ small compared with $x_{0}$ the approximate formula holds

\sqrt[n]{x_{0}+\Delta x} \approx \sqrt[n]{x_{0}}+\frac{\sqrt[n]{x_{0}}}{n x_{0}} \Delta x, \quad x_{0} > 0

Using this formula, approximately compute:

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(1)

$\sqrt{640}$

(2)

$\sqrt[3]{200}$

(3)

$\sqrt[5]{243,45}$

(4)

$\sqrt[10]{1000}$

Problem 1.13.220

Determine by approximately how much the volume of a ball increases if its radius $R = 15$ cm is increased by 0.2 cm.

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Problem 1.13.221

Determine approximately the relative error in computing the surface area of a sphere if the relative error in determining its radius was $1\%$ .

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Problem 1.13.222

By approximately how much (in percent) will the current in a conductor change if its resistance increases by $1\%$ ?

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Problem 1.13.223

By approximately how much should the length of a pendulum $l = 20 \mathrm{~ cm}$ be changed so that the period of oscillation of the pendulum increases by $0.05 \mathrm{s}$ ? The period $T$ is given by the formula $T = 2 \pi \sqrt{l / g}$ .

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