derivative f(x)=5xsqrt(1-x^3)

解

導関数

f (x) = 5 x 1 - x^{3}

\frac{5 ( - 5 x ^{3} + 2 )}{2 1 - x ^{3}}

解答ステップ

\frac{d}{d x} (5 x 1 - x^{3})

定数を除去:

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

= 5 \frac{d}{d x} (x 1 - x^{3})

積の法則を適用:

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

f = x, g = 1 - x^{3}

= 5 (\frac{d x}{d x} 1 - x^{3} + \frac{d}{d x} (1 - x^{3}) x)

\frac{d x}{d x} = 1

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

= 5 (1 \cdot 1 - x^{3} + (- \frac{3 x ^{2}}{2 1 - x ^{3}}) x)

簡素化

5 (1 \cdot 1 - x^{3} + (- \frac{3 x ^{2}}{2 1 - x ^{3}}) x) : \frac{5 ( - 5 x ^{3} + 2 )}{2 1 - x ^{3}}

= \frac{5 ( - 5 x ^{3} + 2 )}{2 1 - x ^{3}}