derivative of (x^3+2x^{1/2}(x^{1/3}+5))

解

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

\frac{10 x ^{3} + 5 x ^{\frac{1}{2}} + 15 x ^{\frac{1}{6}} + 45 x ^{\frac{8}{3}}}{3 x ^{\frac{2}{3}}}

解答ステップ

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

積の法則を適用:

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

f = x^{3} + 2 x^{\frac{1}{2}}, g = x^{\frac{1}{3}} + 5

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

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

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

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

簡素化

\frac{1 + 3 x ^{\frac{5}{2}}}{x ^{\frac{1}{2}}} (x^{\frac{1}{3}} + 5) + \frac{1}{3 x ^{\frac{2}{3}}} (x^{3} + 2 x^{\frac{1}{2}}) : \frac{10 x ^{3} + 5 x ^{\frac{1}{2}} + 15 x ^{\frac{1}{6}} + 45 x ^{\frac{8}{3}}}{3 x ^{\frac{2}{3}}}

= \frac{10 x ^{3} + 5 x ^{\frac{1}{2}} + 15 x ^{\frac{1}{6}} + 45 x ^{\frac{8}{3}}}{3 x ^{\frac{2}{3}}}