Question

Требуется найти производную 1) y= ($\sqrt[3]{5}$ - 4$\sqrt[8]{7}$ )'
2) (tg$\sqrt{x}$ - 7x^5 +3cos45°)'
3)(e^x +3cos$\frac{1}{3}$ x)'
4)(5x^6 -2x^4 + $\frac{3}{x^5}$)'
5) ($\sqrt[3]{x^7}$ -2$\sqrt[4]{x}$ +$\sqrt{3}$ )'
6) ($\frac{cos x +3x}{sinx}$ ) '
7) (tg x * $\sqrt{e^x}$)'
8)($\frac{x}{sin\sqrt{x} }$)'
9)($\frac{x^2+3x}{x(^3)-1 }$)'

Answer 1

1

$y = \sqrt[3]{5} - 4 \sqrt[8]{7} \\ y' = 0$

2

$y = tg( \sqrt{x} ) - 7 {x}^{5} + 3 \cos(45^{\circ} )$

$y '= \frac{1}{ \cos {}^{2} ( \sqrt{x} ) } \times ( \sqrt{x} ) '- 35 {x}^{4} + 0 = \\ = \frac{1}{ \cos {}^{2} ( \sqrt{x} ) } \times \frac{1}{2} {x}^{ - \frac{1}{2} } - 35 {x}^{4} = \frac{1}{2 \sqrt{x} \cos {}^{2} ( \sqrt{x} ) } - 35 {x}^{4}$

3

$(e {}^{x} + 3 \cos( \frac{x}{3} ) )' = {e}^{x} - 3 \sin( \frac{x}{3} ) \times ( \frac{x}{3} )' = \\ = {e}^{2} - 3 \sin( \frac{x}{3} ) \times \frac{1}{3} = e {}^{x} - \sin( \frac{x}{3} )$

4

$(5 {x}^{6} - 2 {x}^{4} + \frac{3}{ {x}^{5} } ) '= (5 {x}^{6} - 2 {x}^{4} + 3 {x}^{ - 5} )' = \\ = 30 {x}^{5} - 8 {x}^{3} - 15 {x}^{ - 6} = 30 {x}^{5} - 8 {x}^{3} - \frac{15}{ {x}^{6} }$

5

$( \sqrt[3]{ {x}^{7} } - 2 \sqrt[4]{x} + \sqrt{3} )' = ( {x}^{ \frac{7}{3} } + 2 {x}^{ \frac{1}{4} } + \sqrt{3} ) '= \\ = \frac{7}{3} {x}^{ \frac{4}{3} } + 2 \times \frac{1}{4} {x}^{ - \frac{3}{4} } + 0 = \\ = \frac{7}{3} \sqrt[3]{ {x}^{4} } + \frac{1}{2 \sqrt[4]{ {x}^{3} } }$

6

$( \frac{ \cos(x) + 3x }{ \sin(x) } )' = \frac{( \cos(x) + 3x) '\times \sin(x) - ( \sin(x)) '\times (\cos(x) + 3x)}{ { \sin }^{2} (x)} = \\ = \frac{( - \sin(x) + 3) \times \sin(x) - \cos(x) \times ( \cos(x) + 3x)}{ \sin {}^{2} (x) } = \\ = \frac{ - \sin {}^{2} (x) + 3 \sin(x) - \cos {}^{2} (x) - 3x \cos(x) }{ \sin {}^{2} (x) } = \frac{3 \sin(x) - 3x \cos( x) - 1}{ \sin {}^{2} (x) }$

7

$(tgx \times \sqrt{ {e}^{x} } )' = (tgx)' \times \sqrt{e {}^{x} } + ( {e}^{ \frac{x}{2} })' \times tgx = \\ = \frac{1}{ \cos {}^{2} (x) } \times \sqrt{ {e}^{x} } + \frac{1}{2} {e}^{ \frac{x}{2} } \times tgx = \\ = \frac{ \sqrt{ {e}^{x} } }{ \cos {}^{2} (x) } + \frac{tgx \sqrt{ {e}^{x} } }{2}$

8

$(\frac{x}{ \sin( \sqrt{x} ) } ) '= \frac{x '\sin( \sqrt{x} ) - \sin( \sqrt{x} ) ' \times ( \sqrt{x}) \times x }{ \sin {}^{2} ( \sqrt{x} ) } = \\ = \frac{ \sin( \sqrt{x} ) - \cos( \sqrt{x} ) \times \frac{1}{2 \sqrt{x} } \times x}{ \sin {}^{2} ( \sqrt{x} ) } = \\ = \frac{ \sin( \sqrt{x} ) - \frac{ \sqrt{x} }{2} \cos( \sqrt{x} ) }{ \sin {}^{2} ( \sqrt{x} ) }$

9

$( \frac{ {x}^{2} + 3x }{ {x}^{3} - 1 } ) '= \frac{( {x}^{2} + 3x)'( {x}^{3} - 1) - ( {x}^{3} - 1)'( {x}^{2} + 3x) }{ {( {x}^{3} - 1)}^{2} } = \\ = \frac{(2x + 3)( {x}^{3} - 1) - 3 {x}^{2} ( {x}^{2} + 3x)}{ {( {x}^{3} - 1)}^{2} } = \\ = \frac{2 {x}^{4} - 2x + 3 {x}^{3} - 3 - 3 {x}^{4} - 9 {x}^{3} }{ {( {x}^{3} - 1) }^{2} } = \\ = \frac{ - {x}^{4} - 6 {x}^{3} - 2x - 3 }{ {( {x}^{3} - 1) }^{2} } = - \frac{ {x}^{4} + 6 {x}^{3} + 2x + 3}{ { ({x}^{3} - 1) }^{2} }$