derivative of x^2(1-xsqrt(x))

解

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

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

解答ステップ

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

簡素化

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

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

積の法則を適用:

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

f = x^{2}, g = 1 - x^{\frac{3}{2}}

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

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

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

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

簡素化

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

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