derivative f(x)=(3+x^5)^3-(2+x^3)^4

解

導関数

f (x) = (3 + x^{5})^{3} - (2 + x^{3})^{4}

15 x^{14} - 12 x^{11} + 90 x^{9} - 72 x^{8} - 144 x^{5} + 135 x^{4} - 96 x^{2}

解答ステップ

\frac{d}{d x} ((3 + x^{5})^{3} - (2 + x^{3})^{4})

簡素化

(3 + x^{5})^{3} - (2 + x^{3})^{4} : x^{15} - x^{12} + 9 x^{10} - 8 x^{9} - 24 x^{6} + 27 x^{5} - 32 x^{3} + 11

= \frac{d}{d x} (x^{15} - x^{12} + 9 x^{10} - 8 x^{9} - 24 x^{6} + 27 x^{5} - 32 x^{3} + 11)

和/差の法則を適用:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (x^{15}) - \frac{d}{d x} (x^{12}) + \frac{d}{d x} (9 x^{10}) - \frac{d}{d x} (8 x^{9}) - \frac{d}{d x} (24 x^{6}) + \frac{d}{d x} (27 x^{5}) - \frac{d}{d x} (32 x^{3}) + \frac{d}{d x} (11)

\frac{d}{d x} (x^{15}) = 15 x^{14}

\frac{d}{d x} (x^{12}) = 12 x^{11}

\frac{d}{d x} (9 x^{10}) = 90 x^{9}

\frac{d}{d x} (8 x^{9}) = 72 x^{8}

\frac{d}{d x} (24 x^{6}) = 144 x^{5}

\frac{d}{d x} (27 x^{5}) = 135 x^{4}

\frac{d}{d x} (32 x^{3}) = 96 x^{2}

\frac{d}{d x} (11) = 0

= 15 x^{14} - 12 x^{11} + 90 x^{9} - 72 x^{8} - 144 x^{5} + 135 x^{4} - 96 x^{2} + 0

簡素化

= 15 x^{14} - 12 x^{11} + 90 x^{9} - 72 x^{8} - 144 x^{5} + 135 x^{4} - 96 x^{2}