Question

Найдите производные следующих функций и Найдите производные следующих начения функций в точке х=1

Answer 1

1

$( {x}^{5} + 1) '= 5 {x}^{4} \\ 5 \times 1 = 5$

2

$( - \frac{1}{x} - 3x)' = ( - {x}^{ - 1} - 3x) '= \\ = {x}^{ - 2} - 3 = \frac{1}{ {x}^{2} } - 3 \\ \frac{1}{1} - 3 = - 2$

3

$(4 {x}^{4} + \sqrt{x} )' = (4 {x}^{4} + {x}^{ \frac{1}{2} } )' = \\ = 16 {x}^{3} + \frac{1}{2} {x}^{ - \frac{1}{2} } = 16 {x}^{3} + \frac{1}{2 \sqrt{x} } \\ 16 + \frac{1}{2} = 16.5$

4

$( \frac{1}{3} {x}^{3} - 2 \sqrt{x} + \frac{5}{x} )' = {x}^{2} - 2 \times \frac{1}{2} {x}^{ - \frac{1}{2} } - 5 {x}^{ - 2} = \\ = {x}^{2} - \frac{1}{ \sqrt{x} } - \frac{5}{ {x}^{2} } \\ 1 - 1 - 5 = - 5$

5

$((5x - 4)(2 {x}^{4} - 7x + 1)) '= \\ = (5x - 4)'(2 {x}^{4} - 7x + 1) + (2 {x}^{4} - 7x + 1)'(5x - 4) = \\ = 5(2 {x}^{4} - 7x + 1) + (8 {x}^{3} - 7)(5x - 4) = \\ = 10 {x}^{4} - 35x + 5 + 40 {x}^{4} - 32 {x}^{3 } - 35x + 28 = \\ = 50 {x}^{4} - 32 {x}^{3} - 70x + 33 \\ 50 - 32 - 70 + 33 = - 20 + 1 = - 19$

6

$( \frac{ {x}^{3} - 7}{3 - 4{x}^{4} } ) '= \frac{( {x}^{3} - 7)'(3 - 4 {x}^{4} ) - (3 - 4 {x}^{4} )'( {x}^{3} - 7) }{ {(3 - 4 {x}^{4} )}^{2} } = \\ = \frac{3 {x}^{2}(3 - 4 {x}^{4}) + 16 {x}^{3} ( {x}^{3} - 7) }{ {(3 - 4 {x}^{4} )}^{2} } = \\ = \frac{9 {x}^{2} - 12 {x}^{6} + 16 {x}^{6} - 112 {x}^{3} }{ {(3 - 4 {x}^{4} )}^{2} } = \frac{4 {x}^{6} - 112 {x}^{3} + 9 {x}^{2} }{ {(3 - 4{x}^{4}) }^{2} } \\ \frac{4 - 112 + 9}{1} = - 109$