derivative of 1/9 (x^2+6^{3/2})

解

\frac{d}{d x} (\frac{1}{9} (x^{2} + 6)^{\frac{3}{2}})

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

解答ステップ

\frac{d}{d x} (\frac{1}{9} (x^{2} + 6)^{\frac{3}{2}})

定数を除去:

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

= \frac{1}{9} \frac{d}{d x} ((x^{2} + 6)^{\frac{3}{2}})

連鎖法則を適用:

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

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

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

= \frac{1}{9} \cdot \frac{3 ( x ^{2} + 6 ) ^{\frac{1}{2}}}{2} \cdot 2 x

簡素化

\frac{1}{9} \cdot \frac{3 ( x ^{2} + 6 ) ^{\frac{1}{2}}}{2} \cdot 2 x : \frac{x ( x ^{2} + 6 ) ^{\frac{1}{2}}}{3}

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