Question

Найдите производную функций
1) y=cos x/ 3x
2)y= 3x^2-4x/5x+2
3) y= (5-cos4x)^3

Answer 1

1)

$y = \frac{ \cos(x) }{3x}$

$y' = \frac{ - \sin( x ) \times 3x - 3 \cos(x) }{ 9 {x}^{2} } = \\ = \frac{ 3(- x \sin(x) - \cos(x)) }{9 {x}^{2} } = \\ = \frac{ - x \sin(x) - \cos(x) }{3 {x}^{2} }$

2)

$y = \frac{3 {x}^{2} - 4x}{5x + 2}$

$y' = \frac{(6x - 4)(5x + 2) - 5(3 {x}^{2} - 4x)}{ {(5x + 2)}^{2} } = \\ = \frac{30 {x}^{2} + 12x - 20x - 15 {x}^{2} + 20x }{ {(5x + 2)}^{2} } = \\ = \frac{15 {x}^{2} + 12x }{ {(5x + 2)}^{2} }$

3)

$y = {(5 - \cos(4x)) }^{3}$

$y' = 3 {(5 - \cos(4x)) }^{2} \times \sin(4x) \times 4 = \\ = 12 \sin(4x) {(5 - \cos(4x)) }^{2}$