1) Найдите производную функции: 1. 1)y=3x⁴; 2) y=2x⁻³; 3) y=4x²; 4) y = ³√x². 2. 1)y=1/x; 2)y=³√x; 3) y = 1/√x; 4) y = 1/⁴√x³.

Краткое пояснение:

Решение:

Задание 1:

• 1. \( y = 3x^4 \)
\( y' = 3 · 4x^{4-1} = 12x^3 \)
• 2. \( y = 2x^{-3} \)
\( y' = 2 · (-3)x^{-3-1} = -6x^{-4} = -\frac{6}{x^4} \)
• 3. \( y = 4x^2 \)
\( y' = 4 · 2x^{2-1} = 8x \)
• 4. \( y = ³√{x^2} = x^{2/3} \)
\( y' = \frac{2}{3}x^{\frac{2}{3}-1} = \frac{2}{3}x^{-\frac{1}{3}} = \frac{2}{3³√{x}} \)

Задание 2:

• 1. \( y = \frac{1}{x} = x^{-1} \)
\( y' = -1x^{-1-1} = -x^{-2} = -\frac{1}{x^2} \)
• 2. \( y = ³√{x} = x^{1/3} \)
\( y' = \frac{1}{3}x^{\frac{1}{3}-1} = \frac{1}{3}x^{-\frac{2}{3}} = \frac{1}{3³√{x^2}} \)
• 3. \( y = \frac{1}{√{x}} = x^{-1/2} \)
\( y' = -\frac{1}{2}x^{-\frac{1}{2}-1} = -\frac{1}{2}x^{-\frac{3}{2}} = -\frac{1}{2x·√{x}} \)
• 4. \( y = \frac{1}{⁴√{x^3}} = x^{-3/4} \)
\( y' = -\frac{3}{4}x^{-\frac{3}{4}-1} = -\frac{3}{4}x^{-\frac{7}{4}} = -\frac{3}{4⁴√{x^7}}} \)

Ответ:

• 1. \( 12x^3, -\frac{6}{x^4}, 8x, \frac{2}{3³√{x}} \)
• 2. \( -\frac{1}{x^2}, \frac{1}{3³√{x^2}}, -\frac{1}{2x·√{x}}, -\frac{3}{4⁴√{x^7}}} \)