Упр.209 ГДЗ Колмогоров 10-11 класс (Алгебра)

Найдём производные по правилу производной произведения.

1. $$f(x)=x^3(4+2x-x^2)$$

   $$f'(x)=(x^3)'(4+2x-x^2)+x^3(4+2x-x^2)’ \\\\ =3x^2(4+2x-x^2)+x^3(2-2x) \\\\ =12x^2+6x^3-3x^4+2x^3-2x^4 \\\\ =-5x^4+8x^3+12x^2$$

2. $$f(x)=\sqrt{x}(2x^2-x)$$

   $$f(x)=x^{\frac12}(2x^2-x)$$

   $$f'(x)=\left(x^{\frac12}\right)'(2x^2-x)+x^{\frac12}(2x^2-x)’ \\\\ =\frac12 x^{-\frac12}(2x^2-x)+x^{\frac12}(4x-1) \\\\ =x^{\frac32}-\frac12 x^{\frac12}+4x^{\frac32}-x^{\frac12} \\\\ =5x^{\frac32}-\frac32 x^{\frac12} \\\\ =\sqrt{x}\left(5x-\frac32\right)$$

3. $$f(x)=x^2(3x+x^3)$$

   $$f'(x)=(x^2)'(3x+x^3)+x^2(3x+x^3)’ \\\\ =2x(3x+x^3)+x^2(3+3x^2) \\\\ =6x^2+2x^4+3x^2+3x^4 \\\\ =9x^2+5x^4$$

4. $$f(x)=(2x-3)(1-x^3)$$

   $$f'(x)=(2x-3)'(1-x^3)+(2x-3)(1-x^3)’ \\\\ =2(1-x^3)-3x^2(2x-3) \\\\ =2-2x^3-6x^3+9x^2 \\\\ =-8x^3+9x^2+2$$

Ответ:

а) $$f'(x)=-5x^4+8x^3+12x^2$$; б) $$f'(x)=\sqrt{x}\left(5x-\frac32\right)$$; в) $$f'(x)=9x^2+5x^4$$; г) $$f'(x)=-8x^3+9x^2+2$$.