derivative (x^2(x^2+9x))^6

解

導関数

(x^{2} (x^{2} + 9 x))^{6}

12 x^{11} (x^{2} + 9 x)^{6} + 6 (x^{2} + 9 x)^{5} (2 x + 9) x^{12}

解答ステップ

\frac{d}{d x} ((x^{2} (x^{2} + 9 x))^{6})

簡素化

(x^{2} (x^{2} + 9 x))^{6} : x^{12} (x^{2} + 9 x)^{6}

= \frac{d}{d x} (x^{12} (x^{2} + 9 x)^{6})

積の法則を適用:

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

f = x^{12}, g = (x^{2} + 9 x)^{6}

= \frac{d}{d x} (x^{12}) (x^{2} + 9 x)^{6} + \frac{d}{d x} ((x^{2} + 9 x)^{6}) x^{12}

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

\frac{d}{d x} ((x^{2} + 9 x)^{6}) = 6 (x^{2} + 9 x)^{5} (2 x + 9)

= 12 x^{11} (x^{2} + 9 x)^{6} + 6 (x^{2} + 9 x)^{5} (2 x + 9) x^{12}