derivative 3x^2(x^3+1)^7

解

導関数

3 x^{2} (x^{3} + 1)^{7}

3 (2 x (x^{3} + 1)^{7} + 21 x^{4} (x^{3} + 1)^{6})

解答ステップ

\frac{d}{d x} (3 x^{2} (x^{3} + 1)^{7})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{d}{d x} (x^{2} (x^{3} + 1)^{7})

積の法則を適用:

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

f = x^{2}, g = (x^{3} + 1)^{7}

= 3 (\frac{d}{d x} (x^{2}) (x^{3} + 1)^{7} + \frac{d}{d x} ((x^{3} + 1)^{7}) x^{2})

\frac{d}{d x} (x^{2}) = 2 x

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

= 3 (2 x (x^{3} + 1)^{7} + 21 x^{2} (x^{3} + 1)^{6} x^{2})

21 x^{2} (x^{3} + 1)^{6} x^{2} = 21 x^{4} (x^{3} + 1)^{6}

= 3 (2 x (x^{3} + 1)^{7} + 21 x^{4} (x^{3} + 1)^{6})