Найдите производную функции: 1) x^(7/9); 2) x^(-9); 3) x^(12/17); 4) x^(-3/14); 5) 1/(4√x^7); 6) 1/x^3; 7) √x; 8) 7√x^5; 9) 1/x; 10) (13x + 5)^4; 11) (8-3x)^(-4); 12) ∬√(-1-9x).

Решение:

1. \( (x^{\frac{7}{9}})' = \frac{7}{9}x^{\frac{7}{9}-1} = \frac{7}{9}x^{-\frac{2}{9}} \)
2. \( (x^{-9})' = -9x^{-9-1} = -9x^{-10} \)
3. \( (x^{\frac{12}{17}})' = \frac{12}{17}x^{\frac{12}{17}-1} = \frac{12}{17}x^{-\frac{5}{17}} \)
4. \( (x^{-\frac{3}{14}})' = -\frac{3}{14}x^{-\frac{3}{14}-1} = -\frac{3}{14}x^{-\frac{17}{14}} \)
5. \( (\frac{1}{4\sqrt{x^7}})' = (\frac{1}{4}x^{-\frac{7}{2}})' = \frac{1}{4} * (-\frac{7}{2})x^{-\frac{7}{2}-1} = -\frac{7}{8}x^{-\frac{9}{2}} \)
6. \( (\frac{1}{x^3})' = (x^{-3})' = -3x^{-3-1} = -3x^{-4} \)
7. \( (\sqrt{x})' = (x^{\frac{1}{2}})' = \frac{1}{2}x^{\frac{1}{2}-1} = \frac{1}{2}x^{-\frac{1}{2}} \)
8. \( (\sqrt[7]{x^5})' = (x^{\frac{5}{7}})' = \frac{5}{7}x^{\frac{5}{7}-1} = \frac{5}{7}x^{-\frac{2}{7}} \)
9. \( (\frac{1}{x})' = (x^{-1})' = -1x^{-1-1} = -x^{-2} \)
10. \( ((13x+5)^4)' = 4(13x+5)^{4-1} * 13 = 52(13x+5)^3 \)
11. \( ((8-3x)^{-4})' = -4(8-3x)^{-4-1} * (-3) = 12(8-3x)^{-5} \)
12. \( (\sqrt[6]{-1-9x})' = ((-1-9x)^{\frac{1}{6}})' = \frac{1}{6}(-1-9x)^{\frac{1}{6}-1} * (-9) = -\frac{3}{2}(-1-9x)^{-\frac{5}{6}} \)