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

解

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

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

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

商の法則を適用:

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

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

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

\frac{\partial}{\partial x} (x^{4} + y^{2}) = 4 x^{3}

= y \frac{2 x ( x ^{4} + y ^{2} ) - 4 x ^{3} x ^{2}}{( x ^{4} + y ^{2} ) ^{2}}

簡素化

y \frac{2 x ( x ^{4} + y ^{2} ) - 4 x ^{3} x ^{2}}{( x ^{4} + y ^{2} ) ^{2}} : \frac{y ( - 2 x ^{5} + 2 x y ^{2} )}{( x ^{4} + y ^{2} ) ^{2}}

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