Question

производная
$y = sin(2x) cos(x)$

$y = frac{ sin(x) }{2 + cos(x) }$

решить:

$frac{2}{ sqrt{7} + sqrt{5} }$

$frac{2x - 2y}{ sqrt{13x} + sqrt{13y} }$

$frac{x}{ sqrt{19} }$

Answer 1

$1); ; y=sin2xcdot cosx\\y'=2, cos2xcdot cosx+sin2xcdot (-sinx)=2, cos2xcdot cosx-sin2xcdot sinx\\2); ; y=frac{sinx}{2+cosx}\\y'=frac{cosx(2+cosx)-sinxcdot (-sinx)}{(2+cosx)^2}=frac{cosx(2+cosx)+sin^2x}{(2+cosx)^2}=frac{2cosx+1}{(2+cosx)^2}\\3); ; frac{2}{sqrt7+sqrt5}=frac{2(sqrt7-sqrt5)}{(sqrt7+sqrt5)(sqrt7-sqrt5)}=frac{2(sqrt7-sqrt5)}{7-5}=sqrt7-sqrt5\\4); ; frac{2x-2y}{sqrt{13x}+sqrt{13y}}=frac{2(x-y)}{sqrt{13}, (sqrt{x}+sqrt{y})}=frac{2(sqrt{x}-sqrt{y})(sqrt{x}+sqrt{y})}{sqrt{13}, (sqrt{x}+sqrt{y})}=frac{2, (sqrt{x}-sqrt{y})}{sqrt{13}}$

$5); ; frac{x}{sqrt{19}}=frac{xsqrt{19}}{19}$