derivative (3(7x^2-2x+6))/(4x^{5/2)}

解

導関数

\frac{3 ( 7 x ^{2} - 2 x + 6 )}{4 x ^{\frac{5}{2}}}

\frac{3 ( - 7 x ^{2} + 6 x - 30 )}{8 x ^{\frac{7}{2}}}

解答ステップ

\frac{d}{d x} (\frac{3 ( 7 x ^{2} - 2 x + 6 )}{4 x ^{\frac{5}{2}}})

定数を除去:

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

= \frac{3}{4} \frac{d}{d x} (\frac{7 x ^{2} - 2 x + 6}{x ^{\frac{5}{2}}})

商の法則を適用:

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

= \frac{3}{4} \cdot \frac{\frac{d}{d x} ( 7 x ^{2} - 2 x + 6 ) x ^{\frac{5}{2}} - \frac{d}{d x} ( x ^{\frac{5}{2}} ) ( 7 x ^{2} - 2 x + 6 )}{( x ^{\frac{5}{2}} ) ^{2}}

\frac{d}{d x} (7 x^{2} - 2 x + 6) = 14 x - 2

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

= \frac{3}{4} \cdot \frac{( 14 x - 2 ) x ^{\frac{5}{2}} - \frac{5 x ^{\frac{3}{2}}}{2} ( 7 x ^{2} - 2 x + 6 )}{( x ^{\frac{5}{2}} ) ^{2}}

簡素化

\frac{3}{4} \cdot \frac{( 14 x - 2 ) x ^{\frac{5}{2}} - \frac{5 x ^{\frac{3}{2}}}{2} ( 7 x ^{2} - 2 x + 6 )}{( x ^{\frac{5}{2}} ) ^{2}} : \frac{3 ( - 7 x ^{2} + 6 x - 30 )}{8 x ^{\frac{7}{2}}}

= \frac{3 ( - 7 x ^{2} + 6 x - 30 )}{8 x ^{\frac{7}{2}}}