Question

Допоможіть будь ласка срочно

Answer 1

1.

$1) \: f(x) = {x}^{5} \\ f'(x) = 5 {x}^{5 - 1} = 5 {x}^{4}$

$2) \: f(x) = 7 \sin(x) \\ f'(x) = 7 \cos(x)$

$3) \: q(x) = 5 {x}^{ - 3} \\ q'(x) = - 3 \times 5 {x}^{ - 3 - 1} = - 15 x{}^{ - 4} = - \frac{15}{ {x}^{4} }$

$4) \: q(x) = \tg(x) \\ q'(x) = \frac{1}{ \cos {}^{2} (x) }$

2.

$1) \: f(x) = x \cos(x) \\ f'(x) = x' \cos(x) + x( \cos(x) )' = \\ = \cos(x) + x \times ( - \sin(x) ) = \\ = \cos(x) - x \sin(x)$

$2) \: g(x) = \frac{x {}^{2} + 1 }{x - 2} \\ g'(x) = \frac{( {x}^{2} + 1)'(x - 2) - (x - 2)'(x {}^{2} + 1) }{(x -2 ) {}^{2} } = \\ = \frac{2x(x - 2) - ( {x}^{2} + 1)}{(x - 2) {}^{2} } = \\ = \frac{2 {x}^{2} - 4x - {x}^{2} - 1}{(x - 2) {}^{2} } = \frac{ {x}^{2} - 4x - 1 }{(x - 2) {}^{2} }$