derivative (x^{63}-1)/(x-1)

解答

导数

\frac{x ^{63} - 1}{x - 1}

62 x^{61} + 61 x^{60} + 60 x^{59} + 59 x^{58} + 58 x^{57} + 57 x^{56} + 56 x^{55} + 55 x^{54} + 54 x^{53} + 53 x^{52} + 52 x^{51} + 51 x^{50} + 50 x^{49} + 49 x^{48} + 48 x^{47} + 47 x^{46} + 46 x^{45} + 45 x^{44} + 44 x^{43} + 43 x^{42} + 42 x^{41} + 41 x^{40} + 40 x^{39} + 39 x^{38} + 38 x^{37} + 37 x^{36} + 36 x^{35} + 35 x^{34} + 34 x^{33} + 33 x^{32} + 32 x^{31} + 31 x^{30} + 30 x^{29} + 29 x^{28} + 28 x^{27} + 27 x^{26} + 26 x^{25} + 25 x^{24} + 24 x^{23} + 23 x^{22} + 22 x^{21} + 21 x^{20} + 20 x^{19} + 19 x^{18} + 18 x^{17} + 17 x^{16} + 16 x^{15} + 15 x^{14} + 14 x^{13} + 13 x^{12} + 12 x^{11} + 11 x^{10} + 10 x^{9} + 9 x^{8} + 8 x^{7} + 7 x^{6} + 6 x^{5} + 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1

求解步骤

\frac{d}{d x} (\frac{x ^{63} - 1}{x - 1})

化简

\frac{x ^{63} - 1}{x - 1} : (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1) (x^{14} + x^{7} + 1) (x^{42} + x^{21} + 1)

= \frac{d}{d x} ((x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1) (x^{14} + x^{7} + 1) (x^{42} + x^{21} + 1))

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1, g = (x^{14} + x^{7} + 1) (x^{42} + x^{21} + 1)

= \frac{d}{d x} (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1) (x^{14} + x^{7} + 1) (x^{42} + x^{21} + 1) + \frac{d}{d x} ((x^{14} + x^{7} + 1) (x^{42} + x^{21} + 1)) (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)

\frac{d}{d x} (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1) = 6 x^{5} + 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1

\frac{d}{d x} ((x^{14} + x^{7} + 1) (x^{42} + x^{21} + 1)) = 56 x^{55} + 49 x^{48} + 42 x^{41} + 35 x^{34} + 28 x^{27} + 21 x^{20} + 14 x^{13} + 7 x^{6}

= (6 x^{5} + 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1) (x^{14} + x^{7} + 1) (x^{42} + x^{21} + 1) + (56 x^{55} + 49 x^{48} + 42 x^{41} + 35 x^{34} + 28 x^{27} + 21 x^{20} + 14 x^{13} + 7 x^{6}) (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1)

化简

(6 x^{5} + 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1) (x^{14} + x^{7} + 1) (x^{42} + x^{21} + 1) + (56 x^{55} + 49 x^{48} + 42 x^{41} + 35 x^{34} + 28 x^{27} + 21 x^{20} + 14 x^{13} + 7 x^{6}) (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1) : 62 x^{61} + 61 x^{60} + 60 x^{59} + 59 x^{58} + 58 x^{57} + 57 x^{56} + 56 x^{55} + 55 x^{54} + 54 x^{53} + 53 x^{52} + 52 x^{51} + 51 x^{50} + 50 x^{49} + 49 x^{48} + 48 x^{47} + 47 x^{46} + 46 x^{45} + 45 x^{44} + 44 x^{43} + 43 x^{42} + 42 x^{41} + 41 x^{40} + 40 x^{39} + 39 x^{38} + 38 x^{37} + 37 x^{36} + 36 x^{35} + 35 x^{34} + 34 x^{33} + 33 x^{32} + 32 x^{31} + 31 x^{30} + 30 x^{29} + 29 x^{28} + 28 x^{27} + 27 x^{26} + 26 x^{25} + 25 x^{24} + 24 x^{23} + 23 x^{22} + 22 x^{21} + 21 x^{20} + 20 x^{19} + 19 x^{18} + 18 x^{17} + 17 x^{16} + 16 x^{15} + 15 x^{14} + 14 x^{13} + 13 x^{12} + 12 x^{11} + 11 x^{10} + 10 x^{9} + 9 x^{8} + 8 x^{7} + 7 x^{6} + 6 x^{5} + 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1

= 62 x^{61} + 61 x^{60} + 60 x^{59} + 59 x^{58} + 58 x^{57} + 57 x^{56} + 56 x^{55} + 55 x^{54} + 54 x^{53} + 53 x^{52} + 52 x^{51} + 51 x^{50} + 50 x^{49} + 49 x^{48} + 48 x^{47} + 47 x^{46} + 46 x^{45} + 45 x^{44} + 44 x^{43} + 43 x^{42} + 42 x^{41} + 41 x^{40} + 40 x^{39} + 39 x^{38} + 38 x^{37} + 37 x^{36} + 36 x^{35} + 35 x^{34} + 34 x^{33} + 33 x^{32} + 32 x^{31} + 31 x^{30} + 30 x^{29} + 29 x^{28} + 28 x^{27} + 27 x^{26} + 26 x^{25} + 25 x^{24} + 24 x^{23} + 23 x^{22} + 22 x^{21} + 21 x^{20} + 20 x^{19} + 19 x^{18} + 18 x^{17} + 17 x^{16} + 16 x^{15} + 15 x^{14} + 14 x^{13} + 13 x^{12} + 12 x^{11} + 11 x^{10} + 10 x^{9} + 9 x^{8} + 8 x^{7} + 7 x^{6} + 6 x^{5} + 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1

\frac{d}{d x} (\frac{3}{3 x + 1})

x \to \infty lim (\frac{1}{e})

d er i v a t i v e y = e^{x^{3}}

\frac{\partial}{\partial x} (arcsin (x y))

d er i v a t i v e x^{2} sin^{3} (3 x)