(\partial)/(\partial x)((x^2-2y)/(4x^2+2y))

解

\frac{\partial}{\partial x} (\frac{x ^{2} - 2 y}{4 x ^{2} + 2 y})

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

解答ステップ

\frac{\partial}{\partial x} (\frac{x ^{2} - 2 y}{4 x ^{2} + 2 y})

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} ( x ^{2} - 2 y ) ( 4 x ^{2} + 2 y ) - \frac{\partial}{\partial x} ( 4 x ^{2} + 2 y ) ( x ^{2} - 2 y )}{( 4 x ^{2} + 2 y ) ^{2}}

\frac{\partial}{\partial x} (x^{2} - 2 y) = 2 x

\frac{\partial}{\partial x} (4 x^{2} + 2 y) = 8 x

= \frac{2 x ( 4 x ^{2} + 2 y ) - 8 x ( x ^{2} - 2 y )}{( 4 x ^{2} + 2 y ) ^{2}}

簡素化

\frac{2 x ( 4 x ^{2} + 2 y ) - 8 x ( x ^{2} - 2 y )}{( 4 x ^{2} + 2 y ) ^{2}} : \frac{5 x y}{( 2 x ^{2} + y ) ^{2}}

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