derivative of (9x/((x^2+81)^{3/2)})

解

\frac{d}{d x} (\frac{9 x}{( x ^{2} + 81 ) ^{\frac{3}{2}}})

\frac{9 ( - 2 x ^{2} + 81 )}{( x ^{2} + 81 ) ^{\frac{5}{2}}}

解答ステップ

\frac{d}{d x} (\frac{9 x}{( x ^{2} + 81 ) ^{\frac{3}{2}}})

定数を除去:

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

= 9 \frac{d}{d x} (\frac{x}{( x ^{2} + 81 ) ^{\frac{3}{2}}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= 9 \cdot \frac{\frac{d x}{d x} ( x ^{2} + 81 ) ^{\frac{3}{2}} - \frac{d}{d x} ( ( x ^{2} + 81 ) ^{\frac{3}{2}} ) x}{( ( x ^{2} + 81 ) ^{\frac{3}{2}} ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} ((x^{2} + 81)^{\frac{3}{2}}) = 3 x (x^{2} + 81)^{\frac{1}{2}}

= 9 \cdot \frac{1 \cdot ( x ^{2} + 81 ) ^{\frac{3}{2}} - 3 x ( x ^{2} + 81 ) ^{\frac{1}{2}} x}{( ( x ^{2} + 81 ) ^{\frac{3}{2}} ) ^{2}}

簡素化

9 \cdot \frac{1 \cdot ( x ^{2} + 81 ) ^{\frac{3}{2}} - 3 x ( x ^{2} + 81 ) ^{\frac{1}{2}} x}{( ( x ^{2} + 81 ) ^{\frac{3}{2}} ) ^{2}} : \frac{9 ( - 2 x ^{2} + 81 )}{( x ^{2} + 81 ) ^{\frac{5}{2}}}

= \frac{9 ( - 2 x ^{2} + 81 )}{( x ^{2} + 81 ) ^{\frac{5}{2}}}