derivative ((sqrt(x))/3+9)^{3/2}

解

導関数

(\frac{x}{3} + 9)^{\frac{3}{2}}

\frac{( x + 27 ) ^{\frac{1}{2}}}{4 \cdot 3 ^{\frac{1}{2}} x ^{\frac{1}{2}}}

解答ステップ

\frac{d}{d x} ((\frac{x}{3} + 9)^{\frac{3}{2}})

簡素化

(\frac{x}{3} + 9)^{\frac{3}{2}} : (\frac{x + 27}{3})^{\frac{3}{2}}

= \frac{d}{d x} ((\frac{x + 27}{3})^{\frac{3}{2}})

連鎖法則を適用:

\frac{3 ( \frac{x + 27}{3} ) ^{\frac{1}{2}}}{2} \frac{d}{d x} (\frac{x + 27}{3})

= \frac{3 ( \frac{x + 27}{3} ) ^{\frac{1}{2}}}{2} \frac{d}{d x} (\frac{x + 27}{3})

\frac{d}{d x} (\frac{x + 27}{3}) = \frac{1}{6 x ^{\frac{1}{2}}}

= \frac{3 ( \frac{x + 27}{3} ) ^{\frac{1}{2}}}{2} \cdot \frac{1}{6 x ^{\frac{1}{2}}}

簡素化

\frac{3 ( \frac{x + 27}{3} ) ^{\frac{1}{2}}}{2} \cdot \frac{1}{6 x ^{\frac{1}{2}}} : \frac{( x + 27 ) ^{\frac{1}{2}}}{4 \cdot 3 ^{\frac{1}{2}} x ^{\frac{1}{2}}}

= \frac{( x + 27 ) ^{\frac{1}{2}}}{4 \cdot 3 ^{\frac{1}{2}} x ^{\frac{1}{2}}}