М-11 СР «Производная степенной функции» Вариант №1 Найдите производную функции 1)x^4/3, 2)x^-7, 3)x^(9/11), 4)x^(-8/15), 5)1/x^(1/3), 6)√x, 7)1/x^6, 8)∛x^2, 9)1/∛x

Краткое пояснение:

Пошаговое решение:

• 1) \( \frac{d}{dx}(x^{\frac{4}{3}}) = \frac{4}{3}x^{\frac{4}{3}-1} = \frac{4}{3}x^{\frac{1}{3}} \)
• 2) \( \frac{d}{dx}(x^{-7}) = -7x^{-7-1} = -7x^{-8} \)
• 3) \( \frac{d}{dx}(x^{\frac{9}{11}}) = \frac{9}{11}x^{\frac{9}{11}-1} = \frac{9}{11}x^{-\frac{2}{11}} \)
• 4) \( \frac{d}{dx}(x^{-\frac{8}{15}}) = -\frac{8}{15}x^{-\frac{8}{15}-1} = -\frac{8}{15}x^{-\frac{23}{15}} \)
• 5) \( \frac{d}{dx}(x^{-\frac{1}{3}}) = -\frac{1}{3}x^{-\frac{1}{3}-1} = -\frac{1}{3}x^{-\frac{4}{3}} \)
• 6) \( \frac{d}{dx}(\sqrt{x}) = \frac{d}{dx}(x^{\frac{1}{2}}) = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}} = \frac{1}{2\sqrt{x}} \)
• 7) \( \frac{d}{dx}(x^{-6}) = -6x^{-6-1} = -6x^{-7} \)
• 8) \( \frac{d}{dx}(\sqrt[3]{x^2}) = \frac{d}{dx}(x^{\frac{2}{3}}) = \frac{2}{3}x^{\frac{2}{3}-1} = \frac{2}{3}x^{-\frac{1}{3}} \)
• 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) \( \frac{4}{3}x^{\frac{1}{3}} \) 2) \( -7x^{-8} \) 3) \( \frac{9}{11}x^{-\frac{2}{11}} \) 4) \( -\frac{8}{15}x^{-\frac{23}{15}} \) 5) \( -\frac{1}{3}x^{-\frac{4}{3}} \) 6) \( \frac{1}{2\sqrt{x}} \) 7) \( -6x^{-7} \) 8) \( \frac{2}{3}x^{-\frac{1}{3}} \) 9) \( -\frac{1}{3}x^{-\frac{4}{3}} \)