Question

Используя таблицу производных и основные правила дифференцирования, найти производные функций

Answer 1

Ответ:

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

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

$3)y' = ( {x}^{ \frac{11}{5 } } - 3 {x}^{ \frac{ - 3}{2} } + 5 {x}^{ - \frac{2}{3} } )' = \frac{11}{5} {x}^{ \frac{6}{5} } + \frac{9}{2} {x}^{ \frac{ - 5}{2} } - \frac{10}{3} {x}^{ - \frac{5}{3} } = \frac{11}{5} x \sqrt[5]{x} + \frac{9}{2 {x}^{2} \sqrt{x} } - \frac{10}{3x \sqrt[3]{ {x}^{2} } }$

$4)y' = - 2t \times \cos(t) - \sin(t) \times (2 - {t}^{2} ) + 2 \sin(t) + 2t \cos(t) = - 2 \sin(t) + {t}^{2} \sin(t) + 2 \sin(t) = {t}^{2} \sin(t)$

$5)s' \: = - \frac{4}{ \sqrt{3} } t - 4 {(1 + 5t)}^{ - 2} \times 5 = - \frac{4}{ \sqrt{3} } t - \frac{20}{ {(5t + 1)}^{2} }$

$6)z' = \frac{3(9 - {x}^{2}) - ( - 2x) \times 3x }{ {(9 - {x}^{2} )}^{2} } = \frac{27 - 3 {x}^{2} + 6 {x}^{2} }{ {(9 - {x}^{2}) }^{2} } = \frac{3 {x}^{2} + 27}{ {(9 - {x}^{2}) }^{2} }$

$7)y' = \frac{1( \sin(x) + \cos(x)) - ( \cos(x) - \sin(x))x }{ {( \sin(x) + \cos(x)) }^{2} } \\ y( \frac{\pi}{2} ) = \frac{ \sin( \frac{\pi}{2} ) + \cos( \frac{\pi}{2} ) - \frac{\pi}{2}( \cos( \frac{\pi}{2} ) - \sin( \frac{\pi}{2} )) }{ {( \sin( \frac{\pi}{2} ) + \cos( \frac{\pi}{2}))^{2} } } = \frac{1 + 0 + \frac{\pi}{2} }{1} = \frac{3\pi}{2}$

$8)y' = arctg( \frac{1}{2} ) \times ( {(arctgx)}^{ - 1} )' = - arctg( \frac{1}{2} ) \times ( {arctg(x))}^{ - 2} \times \frac{1}{1 + {x}^{2} } = \frac{ - arctg( \frac{1}{2} )}{(1 + {x)}^{2} \times {arctg(x)}^{2} }$

$9)y' = log_{10}(x) + \frac{x}{ ln(10) \times x} = log_{10}(x) + \frac{1}{ ln(10) } = log_{10}(x) + log_{10}(e) = log_{10}(x \times e) = lg(x \times e)$

$10)y' = \frac{1}{2 \sqrt{x} } ln(x) + \frac{1}{x} \sqrt{x} + ln(3) \times {3}^{x} \times {x}^{3} + 3 {x}^{2} \times {3}^{x} = \frac{ ln(x) }{2 \sqrt{x} } + \frac{1}{ \sqrt{x} } + {3}^{x} {x}^{2} ( ln(3) x + 3)$

$11)y' = {e}^{x} arcsin(x) + \frac{1}{ \sqrt{1 - {x}^{2} } } \times {e}^{x}$

$12)y' = - { (\cos(x) )}^{ - 2} \times ( - \sin(x) ) + \frac{1}{2} \cos(x) = \frac{ \sin(x) }{ {( \cos(x) )}^{2} } + \frac{1}{2} \cos(x)$

$13)y' = ln(2) \times {2}^{x} log_{2}(x) + \frac{1}{ ln(2) \times x} \times {2}^{x} = {2}^{x} ( ln(2) \times log_{2}(x) + \frac{1}{x ln(2) } )$