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

解

導関数

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

5 x^{4} - 12 x^{2}

解答ステップ

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

積の法則を適用:

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

f = x^{3}, g = x^{2} - 4

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

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

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

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

簡素化

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

= 5 x^{4} - 12 x^{2}