М-11 СР «Производная степенной функции» Вариант №2 Найдите производную функции 1)x^-7, 2)x^-9, 3)x^(12/17), 4)x^(-3/14), 5)1/√x^7, 6)1/x^3, 7)√x, 8)⁷√x^5, 9)1/x^(1/3)

Краткое пояснение:

Пошаговое решение:

• 1) \( \frac{d}{dx}(x^{-7}) = -7x^{-7-1} = -7x^{-8} \)
• 2) \( \frac{d}{dx}(x^{-9}) = -9x^{-9-1} = -9x^{-10} \)
• 3) \( \frac{d}{dx}(x^{\frac{12}{17}}) = \frac{12}{17}x^{\frac{12}{17}-1} = \frac{12}{17}x^{-\frac{5}{17}} \)
• 4) \( \frac{d}{dx}(x^{-\frac{3}{14}}) = -\frac{3}{14}x^{-\frac{3}{14}-1} = -\frac{3}{14}x^{-\frac{17}{14}} \)
• 5) \( \frac{d}{dx}(\frac{1}{\sqrt{x^7}}) = \frac{d}{dx}(x^{-\frac{7}{2}}) = -\frac{7}{2}x^{-\frac{7}{2}-1} = -\frac{7}{2}x^{-\frac{9}{2}} \)
• 6) \( \frac{d}{dx}(x^{-3}) = -3x^{-3-1} = -3x^{-4} \)
• 7) \( \frac{d}{dx}(\sqrt{x}) = \frac{d}{dx}(x^{\frac{1}{2}}) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}} \)
• 8) \( \frac{d}{dx}(\sqrt[7]{x^5}) = \frac{d}{dx}(x^{\frac{5}{7}}) = \frac{5}{7}x^{\frac{5}{7}-1} = \frac{5}{7}x^{-\frac{2}{7}} \)
• 9) \( \frac{d}{dx}(x^{-\frac{1}{3}}) = -\frac{1}{3}x^{-\frac{1}{3}-1} = -\frac{1}{3}x^{-\frac{4}{3}} \)

Ответ: 1) \( -7x^{-8} \) 2) \( -9x^{-10} \) 3) \( \frac{12}{17}x^{-\frac{5}{17}} \) 4) \( -\frac{3}{14}x^{-\frac{17}{14}} \) 5) \( -\frac{7}{2}x^{-\frac{9}{2}} \) 6) \( -3x^{-4} \) 7) \( \frac{1}{2}x^{-\frac{1}{2}} \) 8) \( \frac{5}{7}x^{-\frac{2}{7}} \) 9) \( -\frac{1}{3}x^{-\frac{4}{3}} \)