Упр.215 ГДЗ Колмогоров 10-11 класс (Алгебра)

Найдём производные функций по правилам дифференцирования частного и произведения.

1. $$f(x)=\frac{x^3-3x}{1+4x^5}$$
   $$f'(x)=\frac{(x^3-3x)'(1+4x^5)-(x^3-3x)(1+4x^5)’}{(1+4x^5)^2}$$
   $$f'(x)=\frac{(3x^2-3)(1+4x^5)-(x^3-3x)\cdot 20x^4}{(1+4x^5)^2}$$
   $$f'(x)=\frac{3x^2+12x^7-3-12x^5-20x^7+60x^5}{(1+4x^5)^2}$$
   $$f'(x)=\frac{-8x^7+48x^5+3x^2-3}{(1+4x^5)^2}$$
2. $$f(x)=\left(\frac{3}{x}+x^2\right)(2-\sqrt{x})$$
   $$f(x)=\left(3x^{-1}+x^2\right)\left(2-x^{1/2}\right)$$
   $$f'(x)=\left(3x^{-1}+x^2\right)'(2-x^{1/2})+\left(3x^{-1}+x^2\right)(2-x^{1/2})’$$
   $$f'(x)=(-3x^{-2}+2x)(2-x^{1/2})+\left(3x^{-1}+x^2\right)\left(-\frac12 x^{-1/2}\right)$$
   $$f'(x)=-6x^{-2}+3x^{-3/2}+4x-2x^{3/2}-\frac32 x^{-3/2}-\frac12 x^{3/2}$$
   $$f'(x)=-6x^{-2}+\frac32 x^{-3/2}+4x-\frac52 x^{3/2}$$
3. $$f(x)=\frac{5-2x^6}{1-x^3}$$
   $$f'(x)=\frac{(5-2x^6)'(1-x^3)-(5-2x^6)(1-x^3)’}{(1-x^3)^2}$$
   $$f'(x)=\frac{(-12x^5)(1-x^3)-(5-2x^6)(-3x^2)}{(1-x^3)^2}$$
   $$f'(x)=\frac{-12x^5+12x^8+15x^2-6x^8}{(1-x^3)^2}$$
   $$f'(x)=\frac{6x^8-12x^5+15x^2}{(1-x^3)^2}$$
4. $$f(x)=\sqrt{x}(3x^5-x)$$
   $$f'(x)=(\sqrt{x})'(3x^5-x)+\sqrt{x}(3x^5-x)’$$
   $$f'(x)=\frac{1}{2\sqrt{x}}(3x^5-x)+\sqrt{x}(15x^4-1)$$
   $$f'(x)=\frac{3x^5-x}{2\sqrt{x}}+15x^{9/2}-x^{1/2}$$
   $$f'(x)=\frac{3}{2}x^{9/2}-\frac12 x^{1/2}+15x^{9/2}-x^{1/2}$$
   $$f'(x)=\frac{33}{2}x^{9/2}-\frac32 x^{1/2}$$
   $$f'(x)=\frac{3}{2}\sqrt{x}\,(11x^4-1)$$

Ответ:

$$\begin{aligned} &\text{а) } f'(x)=\frac{-8x^7+48x^5+3x^2-3}{(1+4x^5)^2},\\ &\text{б) } f'(x)=-6x^{-2}+\frac32 x^{-3/2}+4x-\frac52 x^{3/2},\\ &\text{в) } f'(x)=\frac{6x^8-12x^5+15x^2}{(1-x^3)^2},\\ &\text{г) } f'(x)=\frac{3}{2}\sqrt{x}\,(11x^4-1). \end{aligned}$$