Question

Помгите пожалуйста!!!

Answer 1

$1. displaystyle (x^4+5)'=(x^4)'+5'=bf {4x^3}$
$2. displaystyle (7x- frac{1}{x} )'=(7x)'-( frac{1}{x} )'=7-(- frac{1}{x^2})=bf{7+ frac{1}{x^2} }$
$3. displaystyle (sqrt x+3x^3)'=(sqrt x)'+(3x^3)'=bf{ frac{1}{2 sqrt x} +9x^2}$
$4. displaystyle ( frac{x^3}{5} +3 sqrt x + frac{4}{x})'= ( frac{x^3}{5})'+(3 sqrt x )'+( frac{4}{x})' boxed{=}$
$displaystyle ( frac{x^3}{5} )'= frac{(x^3)' cdot 5- x^3 cdot 5'}{5^2} = frac{3x^2 cdot 5-x^3 cdot 0}{25}= frac{15x^2}{25}= frac{3x^2}{5}$
$displaystyle (3 sqrt x)'=3' cdot sqrt x + 3 cdot (sqrt x)'=0+ 3 cdot frac{1}{2 sqrt x}= frac{3}{2 sqrt x}$
$displaystyle ( frac{4}{x})'= frac{4'x-4x' }{x^2}= -frac{4}{x^2}$
$displaystyle boxed{=}bf{ frac{3x^2}{5}+ frac{3}{2 sqrt x}- frac{4}{x^2}}$
$5. displaystyle ((4x-6)(3x^5+2x- sqrt 5))'=(4x-6)'(3x^5+2x- sqrt 5)+$
$+(4x-6)(3x^5+2x- sqrt 5)'=4(3x^5+2x- sqrt 5)+(4x-6)(15x^4+2)=$
$=12x^5+8x-4 sqrt 5 +60x^5+8x-90x^4-12=bf{72x^5-90x^4+16x-}$
$bf{-12-4 sqrt 5}=2(36x^5-45x^4+8x-6-2 sqrt 5)$
$displaystyle 6. ( frac{x^3+5x}{-4x+x^2} )' = frac{(x^3+5x)'(-4x+x^2)-(x^3+5x)(-4x+x^2)'}{(-4x+x^2)^2 }=$
$displaystyle = frac{(3x^2+5)(-4x+x^2)-(x^3+5x)(-4+2x)}{(-4x+x^2)^2} =$
$displaystyle = frac{-12x^3+3x^4-20x+5x^2-(-4x^3+2x^4-20x+10x^2)}{(-4x+x^2)^2}=$
$displaystyle = frac{-12x^3+3x^4-20x+5x^2+4x^3-2x^4+20x-10x^2}{(-4x+x^2)^2}=$
$displaystyle = frac{x^4-8x^3-5x^2}{x^4-8x^3+16x^2}= frac{x^2(x^2-8x-5)}{x^2(x^2-8x-16)}=bf{ frac{x^2-8x-5}{(x-4)^2} }$