Упр.28.373 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)

1) y=x^6+2x^4+4/x^2-1; 7) y=(2x-1)^6;
2) y=(x^2+x+1)(x^2-4x+1); 8) y=v(x^3-3x);
3) y=(3x-1)/(x^2+1); 9) y=log_3 (2x^2-3x+1);
4) y=(3-2x)vx; 10) y=14^(2-5x);
5) y=vxsin(x); 11) y=x^3+ln (6x-1);
6) y=2^x cos(x); 12) y=3/x-2/x^2+4/x^5.

1. $$y=x^6+2x^4+\frac{4}{x^2}-1$$
   $$y’=6x^5+8x^3-\frac{8}{x^3}$$
2. $$y=(x^2+x+1)(x^2-4x+1)$$
   $$y=x^4-3x^3-2x^2-3x+1$$
   $$y’=4x^3-9x^2-4x-3$$
3. $$y=\frac{3x-1}{x^2+1}$$
   $$y’=\frac{3(x^2+1)-(3x-1)\cdot 2x}{(x^2+1)^2}$$
   $$y’=\frac{3x^2+3-6x^2+2x}{(x^2+1)^2}=\frac{2x-3x^2+3}{(x^2+1)^2}$$
4. $$y=(3-2x)\sqrt{x}$$
   $$y’=-2\sqrt{x}+(3-2x)\cdot \frac{1}{2\sqrt{x}}$$
   $$y’=\frac{-4x+3-2x}{2\sqrt{x}}=\frac{3-6x}{2\sqrt{x}}$$
5. $$y=\sqrt{x}\sin x$$
   $$y’=\frac{1}{2\sqrt{x}}\sin x+\sqrt{x}\cos x$$
   $$y’=\frac{\sin x+2x\cos x}{2\sqrt{x}}$$
6. $$y=2^x\cos x$$
   $$y’=2^x\ln 2\cdot \cos x+2^x(-\sin x)$$
   $$y’=2^x(\ln 2\cos x-\sin x)$$
7. $$y=(2x-1)^6$$
   $$y’=6(2x-1)^5\cdot 2=12(2x-1)^5$$
8. $$y=\sqrt{x^3-3x}$$
   $$y’=\frac{3x^2-3}{2\sqrt{x^3-3x}}$$
9. $$y=\log_3(2x^2-3x+1)$$
   $$y’=\frac{4x-3}{(2x^2-3x+1)\ln 3}$$
10. $$y=14^{\,2-5x}$$
    $$y’=-5\cdot 14^{\,2-5x}\ln 14$$
11. $$y=x^3+\ln(6x-1)$$
    $$y’=3x^2+\frac{6}{6x-1}$$
12. $$y=\frac{3}{x}-\frac{2}{x^2}+\frac{4}{x^5}$$
    $$y’=-\frac{3}{x^2}+\frac{4}{x^3}-\frac{20}{x^6}$$

Ответ

1. $$y’=6x^5+8x^3-\frac{8}{x^3}$$
2. $$y’=4x^3-9x^2-4x-3$$
3. $$y’=\frac{2x-3x^2+3}{(x^2+1)^2}$$
4. $$y’=\frac{3-6x}{2\sqrt{x}}$$
5. $$y’=\frac{\sin x+2x\cos x}{2\sqrt{x}}$$
6. $$y’=2^x(\ln 2\cos x-\sin x)$$
7. $$y’=12(2x-1)^5$$
8. $$y’=\frac{3x^2-3}{2\sqrt{x^3-3x}}$$
9. $$y’=\frac{4x-3}{(2x^2-3x+1)\ln 3}$$
10. $$y’=-5\cdot 14^{\,2-5x}\ln 14$$
11. $$y’=3x^2+\frac{6}{6x-1}$$
12. $$y’=-\frac{3}{x^2}+\frac{4}{x^3}-\frac{20}{x^6}$$