derivative (3t^2+4t-8)/(t^{3/2)}

解

導関数

\frac{3 t ^{2} + 4 t - 8}{t ^{\frac{3}{2}}}

\frac{3 t ^{2} - 4 t + 24}{2 t ^{\frac{5}{2}}}

解答ステップ

\frac{d}{d t} (\frac{3 t ^{2} + 4 t - 8}{t ^{\frac{3}{2}}})

商の法則を適用:

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

= \frac{\frac{d}{d t} ( 3 t ^{2} + 4 t - 8 ) t ^{\frac{3}{2}} - \frac{d}{d t} ( t ^{\frac{3}{2}} ) ( 3 t ^{2} + 4 t - 8 )}{( t ^{\frac{3}{2}} ) ^{2}}

\frac{d}{d t} (3 t^{2} + 4 t - 8) = 6 t + 4

\frac{d}{d t} (t^{\frac{3}{2}}) = \frac{3 t ^{\frac{1}{2}}}{2}

= \frac{( 6 t + 4 ) t ^{\frac{3}{2}} - \frac{3 t ^{\frac{1}{2}}}{2} ( 3 t ^{2} + 4 t - 8 )}{( t ^{\frac{3}{2}} ) ^{2}}

簡素化

\frac{( 6 t + 4 ) t ^{\frac{3}{2}} - \frac{3 t ^{\frac{1}{2}}}{2} ( 3 t ^{2} + 4 t - 8 )}{( t ^{\frac{3}{2}} ) ^{2}} : \frac{3 t ^{2} - 4 t + 24}{2 t ^{\frac{5}{2}}}

= \frac{3 t ^{2} - 4 t + 24}{2 t ^{\frac{5}{2}}}