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

解

導関数

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

3 (8 x^{7} - 12 x^{5})

解答ステップ

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

定数を除去:

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

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

積の法則を適用:

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

f = x^{5}, g = x^{3} - 2 x

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

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

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

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

拡張

5 x^{4} (x^{3} - 2 x) + (3 x^{2} - 2) x^{5} : 8 x^{7} - 12 x^{5}

= 3 (8 x^{7} - 12 x^{5})