derivative of x^4(x^2-2x^2)

解

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

8 x^{7} - 28 x^{6} + 24 x^{5}

解答ステップ

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

積の法則を適用:

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

f = x^{4}, g = (x^{2} - 2 x)^{2}

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

\frac{d}{d x} (x^{4}) = 4 x^{3}

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

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

簡素化

4 x^{3} (x^{2} - 2 x)^{2} + 2 (x^{2} - 2 x) (2 x - 2) x^{4} : 8 x^{7} - 28 x^{6} + 24 x^{5}

= 8 x^{7} - 28 x^{6} + 24 x^{5}