Найдите производную функции 1) y=x²+2x³; 2) y=10x+3√x; 3) y=-+7x; 4) y=tgx+3; 5) y= x²+13x+12; 6) y=(x²+3)⋅(x²-1); 7) y=(x²+1)⋅√x; 8) y=x²⋅cosx; 9) y=(+8)⋅(5x-2); 10) y=\frac{4x-7}{2x+1}; 11) y=\frac{5}{2x²+5}; 12) y=\frac{5sinx}{x}

Ответ:

Разбираемся:

1. \( y = x^2 + 2x^3 \)
   \( y' = 2x + 6x^2 \)
2. \( y = 10x + 3\sqrt{x} = 10x + 3x^{1/2} \)
   \( y' = 10 + \frac{3}{2}x^{-1/2} = 10 + \frac{3}{2\sqrt{x}} \)
3. \( y = -\frac{3}{x} + 7x = -3x^{-1} + 7x \)
   \( y' = 3x^{-2} + 7 = \frac{3}{x^2} + 7 \)
4. \( y = \tan(x) + 3 \)
   \( y' = \sec^2(x) = \frac{1}{\cos^2(x)} \)
5. \( y = x^2 + 13x + 12 \)
   \( y' = 2x + 13 \)
6. \( y = (x^2 + 3)(x^2 - 1) = x^4 + 2x^2 - 3 \)
   \( y' = 4x^3 + 4x \)
7. \( y = (x^2 + 1)\sqrt{x} = (x^2 + 1)x^{1/2} = x^{5/2} + x^{1/2} \)
   \( y' = \frac{5}{2}x^{3/2} + \frac{1}{2}x^{-1/2} = \frac{5}{2}x\sqrt{x} + \frac{1}{2\sqrt{x}} \)
8. \( y = x^2 \cos(x) \)
   \( y' = 2x \cos(x) - x^2 \sin(x) \)
9. \( y = (\frac{1}{x} + 8)(5x - 2) = (x^{-1} + 8)(5x - 2) = 5 - 2x^{-1} + 40x - 16 = 40x - 2x^{-1} - 11 \)
   \( y' = 40 + 2x^{-2} = 40 + \frac{2}{x^2} \)
10. \( y = \frac{4x - 7}{2x + 1} \)
    \( y' = \frac{(4)(2x + 1) - (4x - 7)(2)}{(2x + 1)^2} = \frac{8x + 4 - 8x + 14}{(2x + 1)^2} = \frac{18}{(2x + 1)^2} \)
11. \( y = \frac{5}{2x^2 + 5} \)
    \( y' = \frac{-5(4x)}{(2x^2 + 5)^2} = \frac{-20x}{(2x^2 + 5)^2} \)
12. \( y = \frac{5\sin(x)}{x} \)
    \( y' = \frac{5\cos(x) \cdot x - 5\sin(x) \cdot 1}{x^2} = \frac{5x\cos(x) - 5\sin(x)}{x^2} \)

Ответ: