(\partial)/(\partial x)((5x-y)/(y^2+3x^2))

解

\frac{\partial}{\partial x} (\frac{5 x - y}{y ^{2} + 3 x ^{2}})

\frac{5 y ^{2} - 15 x ^{2} + 6 x y}{( y ^{2} + 3 x ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{5 x - y}{y ^{2} + 3 x ^{2}})

y

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 x} ( 5 x - y ) ( y ^{2} + 3 x ^{2} ) - \frac{\partial}{\partial x} ( y ^{2} + 3 x ^{2} ) ( 5 x - y )}{( y ^{2} + 3 x ^{2} ) ^{2}}

\frac{\partial}{\partial x} (5 x - y) = 5

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

= \frac{5 ( y ^{2} + 3 x ^{2} ) - 6 x ( 5 x - y )}{( y ^{2} + 3 x ^{2} ) ^{2}}

拡張

5 (y^{2} + 3 x^{2}) - 6 x (5 x - y) : 5 y^{2} - 15 x^{2} + 6 x y

= \frac{5 y ^{2} - 15 x ^{2} + 6 x y}{( y ^{2} + 3 x ^{2} ) ^{2}}