Question

Найдите производную функции

Answer 1

1

$( {x}^{3} ) '= 3 {x}^{2}$

2

$( {x}^{5} - \cos(x))' = 5 {x}^{4} - ( - \sin(x) ) = \\ = 5 {x}^{4} + \sin(x)$

3

$(5 \sqrt{x} + 4 ln(x)) ' = (5 {x}^{ \frac{1}{2} } + 4 ln(x))' = \\ = 5 \times \frac{1}{2} {x}^{ - \frac{1}{2} } + \frac{4}{x} = \frac{5}{2 \sqrt{x} } + \frac{4}{x}$

4

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

5

$( \frac{ \cos(x) }{5 {x}^{2} } )' = \frac{( \cos(x))' \times 5 {x}^{2} - (5 {x}^{2} )' \times \cos(x) }{ {(5 {x}^{2} )}^{2} } = \\ = \frac{ - \sin(x) \times 5 {x}^{2} - 10x \cos(x) }{25 {x}^{4} } = - \frac{5x \sin(x) + 10 \cos(x) }{25 {x}^{3} }$

6

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