1) Найдите производную функции: 1. 1) y=3x^4; 2) y = 2x^-3; 3) y=4x^(1/2); 4) y = x^(2/3) 2. 1) y = 1/x^3; 2) y = 3/x^2; 3) y = 1/sqrt(x); 4) y = 1/root(4)(x^3)

Краткое пояснение:

Пошаговое решение:

1. Задание 1:
• 1) \( y = 3x^4 \)
  \( y' = 3 \cdot 4x^{4-1} = 12x^3 \)
• 2) \( y = 2x^{-3} \)
  \( y' = 2 \cdot (-3)x^{-3-1} = -6x^{-4} = -\frac{6}{x^4} \)
• 3) \( y = 4x^{1/2} \)
  \( y' = 4 \cdot \frac{1}{2}x^{1/2-1} = 2x^{-1/2} = \frac{2}{\sqrt{x}} \)
• 4) \( y = x^{2/3} \)
  \( y' = \frac{2}{3}x^{2/3-1} = \frac{2}{3}x^{-1/3} = \frac{2}{3\sqrt[3]{x}} \)
2. Задание 2:
• 1) \( y = \frac{1}{x^3} = x^{-3} \)
  \( y' = -3x^{-3-1} = -3x^{-4} = -\frac{3}{x^4} \)
• 2) \( y = \frac{3}{x^2} = 3x^{-2} \)
  \( y' = 3 \cdot (-2)x^{-2-1} = -6x^{-3} = -\frac{6}{x^3} \)
• 3) \( y = \frac{1}{\sqrt{x}} = x^{-1/2} \)
  \( y' = -\frac{1}{2}x^{-1/2-1} = -\frac{1}{2}x^{-3/2} = -\frac{1}{2\sqrt{x^3}} \)
• 4) \( y = \frac{1}{\sqrt[4]{x^3}} = x^{-3/4} \)
  \( y' = -\frac{3}{4}x^{-3/4-1} = -\frac{3}{4}x^{-7/4} = -\frac{3}{4\sqrt[4]{x^7}} \)

Ответ:

• 1. 1) \(12x^3\); 2) \(-\frac{6}{x^4}\); 3) \(\frac{2}{\sqrt{x}}\); 4) \(\frac{2}{3\sqrt[3]{x}}\).
• 2. 1) \(-\frac{3}{x^4}\); 2) \(-\frac{6}{x^3}\); 3) \(-\frac{1}{2\sqrt{x^3}}\); 4) \(-\frac{3}{4\sqrt[4]{x^7}}\).