Question

Помогите пожалуйста)

Answer 1

$boxed{(uv)'=u'v+uv'}$
$1) y= sqrt{x} cdot 7x^3$
$displaystyle y'= frac{1}{2 sqrt x} cdot 7x^3+ sqrt{x} cdot 21x^2= frac{7x^3}{2 sqrt{x}} +21x^2 sqrt{x}= frac{7x^3+42x^3}{2 sqrt{x}}=$
$displaystyle = frac{49x^3}{2 sqrt{x}}= frac{49x^{7/2}}{2x}= frac{x(49x^{5/2})}{2x}= frac{49x^{5/2}}{2}$

$2) y=cos3x$
$y'=-3sin3x$

$boxed{( frac{u}{v})'= frac{u'v-uv'}{v^2} }$
$displaystyle 3) y= frac{x^6+3x^4}{1-x^2}$
$displaystyle y'= frac{(6x^5+12x^3)(1-x^2)-(x^6+3x^4)(-2x)}{(1-x^2)^2}=$
$displaystyle = frac{6x^5+12x^3-6x^7-12x^5-(-2x^7-6x^5)}{(1-x^2)^2}=$
$displaystyle = frac{6x^5+12x^3-6x^7-12x^5+2x^7+6x^5}{(1-x^2)^2} = frac{12x^3-4x^7}{(1-x^2)^2}$

$boxed{f(g(x))'=f(x)' cdot g(x)'}$
$4) y=arccos sqrt{x-1}$
$displaystyle y'=- frac{1}{sqrt{1-x^2}}cdot sqrt{x-1}+arccos frac{1}{2 sqrt{x-1}}= -frac{sqrt{x-1}}{sqrt{1-x^2}}+frac{arccos}{2sqrt{x-1}}$