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

解

導関数

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

\frac{10 x ^{\frac{3}{2}} - 55 x ^{\frac{9}{2}}}{2}

解答ステップ

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

積の法則を適用:

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

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

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

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

\frac{d}{d x} (2 - 5 x^{3}) = - 15 x^{2}

= \frac{5 x ^{\frac{3}{2}}}{2} (2 - 5 x^{3}) + (- 15 x^{2}) x^{\frac{5}{2}}

簡素化

\frac{5 x ^{\frac{3}{2}}}{2} (2 - 5 x^{3}) + (- 15 x^{2}) x^{\frac{5}{2}} : \frac{10 x ^{\frac{3}{2}} - 55 x ^{\frac{9}{2}}}{2}

= \frac{10 x ^{\frac{3}{2}} - 55 x ^{\frac{9}{2}}}{2}