(\partial)/(\partial s)(s/(sqrt(s^2+t^2)))

解

\frac{\partial}{\partial s} (\frac{s}{s ^{2} + t ^{2}})

\frac{t ^{2}}{( s ^{2} + t ^{2} ) s ^{2} + t ^{2}}

解答ステップ

\frac{\partial}{\partial s} (\frac{s}{s ^{2} + t ^{2}})

t

を定数として扱う

商の法則を適用:

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

= \frac{\frac{\partial}{\partial s} ( s ) s ^{2} + t ^{2} - \frac{\partial}{\partial s} ( s ^{2} + t ^{2} ) s}{( s ^{2} + t ^{2} ) ^{2}}

\frac{\partial}{\partial s} (s) = 1

\frac{\partial}{\partial s} (s^{2} + t^{2}) = \frac{s}{s ^{2} + t ^{2}}

= \frac{1 \cdot s ^{2} + t ^{2} - \frac{s}{s ^{2} + t ^{2}} s}{( s ^{2} + t ^{2} ) ^{2}}

簡素化

\frac{1 \cdot s ^{2} + t ^{2} - \frac{s}{s ^{2} + t ^{2}} s}{( s ^{2} + t ^{2} ) ^{2}} : \frac{t ^{2}}{( s ^{2} + t ^{2} ) s ^{2} + t ^{2}}

= \frac{t ^{2}}{( s ^{2} + t ^{2} ) s ^{2} + t ^{2}}