(\partial)/(\partial x)((4x^3)/(x^2+y^2))

解

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

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

解答ステップ

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

y

を定数として扱う

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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