Упр.820 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\left(2x-3\right)^5\left(3x^2+2x+1\right)$$
2. $$\left(x-1\right)^4\left(x+1\right)^7$$
3. $$\sqrt[4]{3x+2}\left(3x-1\right)^4$$
4. $$\sqrt[3]{2x+1}\left(2x-3\right)^3$$

1) $$f(x)=(2x-3)^5(3x^2+2x+1)$$

Используем правило производной произведения:

$$f'(x)=\bigl((2x-3)^5\bigr)'(3x^2+2x+1)+(2x-3)^5(3x^2+2x+1)’$$

$$f'(x)=5\cdot 2(2x-3)^4(3x^2+2x+1)+(2x-3)^5(6x+2)$$

$$f'(x)=(2x-3)^4\bigl(10(3x^2+2x+1)+(2x-3)(6x+2)\bigr)$$

$$f'(x)=(2x-3)^4(42x^2+6x+4)$$

2) $$f(x)=(x-1)^4(x+1)^7$$

$$f'(x)=\bigl((x-1)^4\bigr)'(x+1)^7+(x-1)^4\bigl((x+1)^7\bigr)’$$

$$f'(x)=4(x-1)^3(x+1)^7+7(x-1)^4(x+1)^6$$

$$f'(x)=(x-1)^3(x+1)^6\bigl(4(x+1)+7(x-1)\bigr)$$

$$f'(x)=(x-1)^3(x+1)^6(11x-3)$$

3) $$f(x)=\sqrt[4]{3x+2}\,(3x-1)^4$$

$$f'(x)=\bigl(\sqrt[4]{3x+2}\bigr)'(3x-1)^4+\sqrt[4]{3x+2}\,\bigl((3x-1)^4\bigr)’$$

$$f'(x)=\frac{3(3x-1)^4}{4\sqrt[4]{(3x+2)^3}}+12\sqrt[4]{3x+2}\,(3x-1)^3$$

$$f'(x)=\frac{3(3x-1)^3\bigl((3x-1)+16(3x+2)\bigr)}{4\sqrt[4]{(3x+2)^3}}$$

$$f'(x)=\frac{3(3x-1)^3(51x+31)}{4\sqrt[4]{(3x+2)^3}}$$

4) $$f(x)=\sqrt[3]{2x+1}\,(2x-3)^3$$

$$f'(x)=\bigl(\sqrt[3]{2x+1}\bigr)'(2x-3)^3+\sqrt[3]{2x+1}\,\bigl((2x-3)^3\bigr)’$$

$$f'(x)=\frac{2(2x-3)^3}{3\sqrt[3]{(2x+1)^2}}+6\sqrt[3]{2x+1}\,(2x-3)^2$$

$$f'(x)=\frac{2(2x-3)^2\bigl((2x-3)+9(2x+1)\bigr)}{3\sqrt[3]{(2x+1)^2}}$$

$$f'(x)=\frac{4(2x-3)^2(10x+3)}{3\sqrt[3]{(2x+1)^2}}$$

Ответ

1) $$f'(x)=(2x-3)^4(42x^2+6x+4)$$

2) $$f'(x)=(x-1)^3(x+1)^6(11x-3)$$

3) $$f'(x)=\frac{3(3x-1)^3(51x+31)}{4\sqrt[4]{(3x+2)^3}}$$

4) $$f'(x)=\frac{4(2x-3)^2(10x+3)}{3\sqrt[3]{(2x+1)^2}}$$