Упр.835 ГДЗ Алимов 10-11 класс (Алгебра)

2) 3 ln x — 2x;
3) log2 x + 1/2x;
4) 3 x^-3 — log3(x);
5) ln (x2 — 2x);
6) (3×2 — 2) log3(x).

1. $$f(x)=2\ln x+3^x$$

   $$f'(x)=(2\ln x)’+(3^x)’=\frac{2}{x}+3^x\ln 3$$

2. $$f(x)=3\ln x-2^x$$

   $$f'(x)=(3\ln x)’-(2^x)’=\frac{3}{x}-2^x\ln 2$$

3. $$f(x)=\log_2 x+\frac{1}{2x}$$

   $$f'(x)=(\log_2 x)’+\left(\frac{1}{2x}\right)’=\frac{1}{x\ln 2}-\frac{1}{2x^2}$$

4. $$f(x)=3x^{-3}-\log_3 x$$

   $$f'(x)=(3x^{-3})’-(\log_3 x)’=3\cdot(-3)x^{-4}-\frac{1}{x\ln 3}=-9x^{-4}-\frac{1}{x\ln 3}$$

5. $$f(x)=\ln(x^2-2x)$$

   Область определения: $$x^2-2x>0$$.

   $$f'(x)=\frac{(x^2-2x)’}{x^2-2x}=\frac{2x-2}{x^2-2x}$$

6. $$f(x)=(3x^2-2)\log_3 x$$

   $$f'(x)=(3x^2-2)’\log_3 x+(3x^2-2)(\log_3 x)’$$

   $$f'(x)=6x\log_3 x+(3x^2-2)\cdot\frac{1}{x\ln 3}$$

   $$f'(x)=\frac{6x^2\ln x+3x^2-2}{x\ln 3}=\frac{3x^2(2\ln x+1)-2}{x\ln 3}$$

Ответ

1. $$\frac{2}{x}+3^x\ln 3$$
2. $$\frac{3}{x}-2^x\ln 2$$
3. $$\frac{1}{x\ln 2}-\frac{1}{2x^2}$$
4. $$-9x^{-4}-\frac{1}{x\ln 3}$$
5. $$\frac{2x-2}{x^2-2x}$$
6. $$\frac{3x^2(2\ln x+1)-2}{x\ln 3}$$