1. Найти производную функции: 1) y = x³/3 - x²/2 2) y = 0,5 x² + 7x - 4 3) y = (3x + 1)⁵ 4) y = √x + 3x⁴ 5) y = sin 6 + e⁴ 6) y = √6x + 1 7) y = eˣ · sin x 8) y = cos(π-x) / sin(2π-x) 9) y = x ln x 10) y = (2x-1) / (x+3) 11) y = 5log₃ x 12) y = 6ˣ

Решение:

1. \( y' = \frac{d}{dx} \left( \frac{x^3}{3} - \frac{x^2}{2} \right) = \frac{1}{3} \cdot 3x^2 - \frac{1}{2} \cdot 2x = x^2 - x \)
2. \( y' = \frac{d}{dx} (0.5 x^2 + 7x - 4) = 0.5 \cdot 2x + 7 = x + 7 \)
3. \( y' = \frac{d}{dx} (3x+1)^5 = 5(3x+1)^4 \cdot 3 = 15(3x+1)^4 \)
4. \( y' = \frac{d}{dx} (x^{1/2} + 3x^4) = \frac{1}{2}x^{-1/2} + 3 \cdot 4x^3 = \frac{1}{2\sqrt{x}} + 12x^3 \)
5. \( y' = \frac{d}{dx} (\sin 6 + e^4) = 0 \) (так как \( \sin 6 \) и \( e^4 \) — константы)
6. \( y' = \frac{d}{dx} (6x+1)^{1/2} = \frac{1}{2}(6x+1)^{-1/2} \cdot 6 = \frac{3}{\sqrt{6x+1}} \)
7. \( y' = \frac{d}{dx} (e^x \cdot \sin x) = e^x \sin x + e^x \cos x = e^x(\sin x + \cos x) \) (используем правило произведения)
8. \( y = \frac{\cos(\pi-x)}{\sin(2\pi-x)} = \frac{-\cos x}{-\sin x} = \cot x \). \( y' = \frac{d}{dx} (\cot x) = -\csc^2 x = -\frac{1}{\sin^2 x} \)
9. \( y' = \frac{d}{dx} (x \ln x) = 1 \cdot \ln x + x \cdot \frac{1}{x} = \ln x + 1 \) (используем правило произведения)
10. \( y' = \frac{d}{dx} \left( \frac{2x-1}{x+3} \right) = \frac{2(x+3) - (2x-1) \cdot 1}{(x+3)^2} = \frac{2x+6 - 2x+1}{(x+3)^2} = \frac{7}{(x+3)^2} \) (используем правило частного)
11. \( y' = \frac{d}{dx} (5\log_3 x) = 5 \cdot \frac{1}{x \ln 3} = \frac{5}{x \ln 3} \)
12. \( y' = \frac{d}{dx} (6^x) = 6^x \ln 6 \)

Ответ: 1) \( x^2 - x \); 2) \( x + 7 \); 3) \( 15(3x+1)^4 \); 4) \( \frac{1}{2\sqrt{x}} + 12x^3 \); 5) \( 0 \); 6) \( \frac{3}{\sqrt{6x+1}} \); 7) \( e^x(\sin x + \cos x) \); 8) \( -\frac{1}{\sin^2 x} \); 9) \( \ln x + 1 \); 10) \( \frac{7}{(x+3)^2} \); 11) \( \frac{5}{x \ln 3} \); 12) \( 6^x \ln 6 \).